Introduction

To further study other types of flow, including those over airfoils and wings, the effects of viscosity must be accounted for. In fact, viscosity plays an essential role in establishing the circulation around a wing through the no-slip condition, boundary-layer development, and the Kutta condition. However, once the circulation is established, much of the lift on an attached wing can be predicted using inviscid flow theory, with viscosity entering primarily through boundary-layer behavior, separation, and drag. In many high-Reynolds-number flows over solid surfaces, the dominant viscous effects are concentrated in a relatively thin region adjacent to the wall, known as the boundary layer. Viscous effects are always significant when fluids flow over solid surfaces or when wakes form behind bodies. The boundary layer is a fundamental concept in aerodynamics and fluid dynamics because it significantly influences a body’s overall aerodynamic characteristics, including lift and drag.

For example, Figure 1 illustrates the boundary layer concept as it develops over the upper and lower surfaces of an airfoil. In this boundary-layer region, viscosity is significant, and what occurs here is critical because it affects the aerodynamic characteristics of the airfoil section and other bodies. However, boundary layers are much thinner in practice than the relative dimensions of the zones shown in Figure 1. Notice that the boundary layer “grows” as it develops downstream, i.e., its thickness (and other properties) change with downstream distance.

One effect of the development of the boundary layer over a body is the creation of shear stresses on its surface, the cumulative manifestation of which is skin-friction drag. Skin friction is often a major source of overall drag on an aircraft, especially for streamlined configurations with large wetted areas. Its contribution depends on the vehicle’s shape, Reynolds number, Mach number, surface conditions, and flight conditions. Another effect is that boundary layers can thicken and detach from surfaces under certain conditions, a phenomenon known as flow separation. The onset of boundary-layer flow separation from a surface generally has a deleterious effect on its aerodynamics, leading to a loss of lift and an increase in drag. In many cases, the aerodynamic characteristics of lifting surfaces and other bodies cannot be understood or predicted without studying the behavior of the boundary layer that develops over them. For this reason, boundary-layer theory is a cornerstone of aerodynamics and is essential for understanding how real aerodynamic surfaces perform in practice.

History

In the seventeenth century, Isaac Newton recognized that fluids resist deformation and that relative motion between adjacent fluid layers gives rise to shear stresses, a relationship now expressed as Newton’s law of viscosity. More than two centuries later, Ludwig Prandtl introduced the concept of the boundary layer. Based on experiments conducted with his students in Germany at the beginning of the twentieth century, Prandtl showed that for flows with relatively small viscosity, the effects of viscous shear are confined primarily to a thin region adjacent to a surface or body, which he termed a boundary layer. This insight was a major advance because it explained how viscosity can strongly influence real flows even when its effects are limited to a relatively small portion of the flow field.

As shown in Figure 2, the flow velocity within a boundary layer increases rapidly from zero at the surface (the no-slip condition) and then asymptotically approaches the external flow velocity, , with increasing distance from the wall. The boundary layer is typically very thin; its characteristic thickness is denoted by . The velocity within the layer, , varies with distance normal to the surface, , so that defines the boundary-layer profile. The form of this profile depends on several factors, including surface condition and the imposed pressure gradient in the external flow.

Boundary layers are broadly classified as laminar or turbulent, each exhibiting distinct velocity distributions and transport characteristics. The behavior of the boundary layer is central to aerodynamics because it determines the wall shear stress and, therefore, the magnitude of skin-friction drag. It also governs whether the flow remains attached to the surface or separates, which has major implications for lift, drag, and overall aerodynamic performance.

By 1910, rapid theoretical and experimental developments established the study of boundary layers as the foundation of modern viscous flow analysis. One of the earliest breakthroughs came from Paul Blasius, a student of Prandtl, who obtained an exact similarity solution for the laminar boundary layer over a flat plate. This result provided the first quantitative predictions of velocity profiles, boundary-layer thickness, and skin-friction drag.

Further advances in the 1920s and 1930s extended the boundary layer theory to more general flows. Theodore von Kármán introduced the integral form of the boundary-layer equations, providing a practical method for estimating boundary-layer growth and drag. At the same time, experimental work began to clarify the transition from laminar to turbulent flow, building on earlier observations by Osborne Reynolds that the character of boundary-layer flow depends on the Reynolds number.

A major complication in understanding and predicting boundary layers is the onset of turbulence. Most flows of practical engineering interest become turbulent after a relatively short distance downstream. Turbulent boundary layers are characterized by stochastic, three-dimensional fluctuations that enhance momentum transport, producing fuller velocity profiles and significantly higher skin-friction drag. Despite its importance, much of turbulent boundary-layer analysis relies on semi-empirical models that continue to be improved and are used in computational fluid dynamics. Predicting transition and modeling turbulence remain among the most challenging problems in fluid mechanics, with direct consequences for drag, separation, noise, and aerodynamic performance.

Laminar & Turbulent Boundary Layers

As detailed in Figure 3, in a laminar boundary layer, the fluid flows in orderly, smooth, unmixed layers; these layers are called laminae, from which the term “laminar” is derived. The fluid layers in a turbulent boundary layer become mixed, so the flow velocities away from the immediate wall region tend to be more uniform because of the fluid’s rotational motion and the resulting shear stresses, vortical eddies, and turbulence. This leads to significant mixing in the flow. For comparable external-flow conditions and downstream distance, turbulent boundary layers are generally thicker than laminar boundary layers because turbulent mixing transports momentum over a larger wall-normal distance.

Another valid quantitative comparison of the profile shapes of a typical laminar and turbulent boundary layer is shown in Figure 4. This presentation is obtained by normalizing the velocity profile by the boundary layer thickness and the outer or edge velocity of the external flow, , i.e., by plotting as a function of . Note that in the case of a turbulent boundary layer, would represent the mean or time-averaged flow velocity, not the instantaneous velocity.

Boundary layer profiles are nearly always plotted in this non-dimensional form because it then allows for ready comparison of their velocity profiles for given values of and , i.e., they are in the general functional form

(1)  

where is the form of the velocity profile. By convention, the edge of the boundary layer is taken to occur at , where . Notice that the velocity gradients, i.e., , in a laminar boundary layer are relatively shallow throughout compared to a turbulent boundary layer.

Because the velocity in the boundary layer smoothly and asymptotically approaches the external flow velocity, the value of must be defined rather carefully to avoid ambiguity. By convention, is defined as the value of for which 99% of the external flow velocity is recovered, i.e., when .

Shear Stresses & Skin Friction

It has already been discussed that viscous stresses in fluids arise whenever there is relative motion between adjacent fluid elements, and these stresses provide a resistance that tends to retard the fluid’s motion. The viscous shear stress, , is related to the absolute viscosity, , by using Newton’s law of viscosity formula, i.e.,

(2)  

where is the rate at which the flow velocity increases (in the direction) and is equivalent to a strain rate, as explained in Figure 5. A fluid must be continuously deformed to sustain a viscous shear stress, whereas an elastic solid can sustain a shear stress under a fixed deformation. Therefore, when the strain rate in a Newtonian fluid is zero, the viscous shear stress is also zero. The strain rate is , so the shallower the value of this gradient, the lower the shear stresses.

As a consequence of the mixing in a turbulent boundary layer, its velocity becomes more uniform further away from the wall. However, this case has steeper velocity gradients as the wall is approached. Because of the flow mixing in a turbulent boundary layer, it develops a greater thickness, , than a laminar boundary layer that forms under the same flow conditions. Equation 2 is a general expression that applies to any point in the flow. However, it is restricted to fluids that follow a linear stress/strain relationship, i.e., a so-called Newtonian fluid, in which the shear stress is proportional to the strain rate. The viscosity is independent of the strain rate, although it may vary with temperature and pressure.

Aerospace engineers are particularly interested in the shear stress produced on the surface or “wall” over which the boundary layer flows, i.e., what happens as , because this value determines the skin friction drag produced by the boundary layer on the surface. The fluid stress at the wall, oriented upstream, induces a reaction shear in the downstream direction. At the wall, the shear stress is

(3)  

or more usually just as

(4)  

As implied by the shape of the boundary layer velocity profiles, as shown in Figure 6, the magnitude of the wall shear stress, , produced in the boundary layer (and hence on the wall) will be greater if a turbulent boundary layer flows over it compared with a laminar one.

This latter result is significant because the boundary layer is the primary source of shear-stress drag, also known as skin-friction drag, on a flight vehicle. Therefore, it would be desirable to minimize drag by maintaining a laminar boundary layer over as much of the vehicle as possible. However, in practice, laminar boundary layers do not persist for long in terms of the downstream distance over which they develop; thus, they transition to turbulence.

Nevertheless, calculating the shear stress from the boundary layer is a prerequisite for determining the drag on any body shape exposed to a flow. As might be expected, drag prediction is challenging for entire aircraft or other flight vehicles because they have complex shapes and three-dimensional flows. In this case, the boundary layers also assume more complex three-dimensional forms.

Development of a Boundary Layer

It should be appreciated that a boundary layer is not a static phenomenon. It is said to “grow” as it develops with downstream distance over a surface. Thus, the thickness and other properties of a boundary layer change continuously as it develops downstream along a given surface. Furthermore, the development of a boundary layer may also be affected by other factors, such as pressure gradients and surface roughness.

A classic example studied by all engineering students is a boundary layer that develops on the top surface of a smooth, flat plate set at zero angle of attack to the oncoming flow, i.e., a uniform-velocity flow. Because the plate is aligned with a uniform external flow, the pressure gradient along its surface is zero. As shown in Figure 7, the thickness of the boundary layer is zero at the leading edge of the plate. Its thickness then grows progressively downstream, with a growth rate that is initially high and then decreases as the boundary layer develops farther along the plate. Note that the vertical scale in Figure 7 has been exaggerated for clarity, and boundary layers are actually very thin. As previously mentioned, the thickness of the boundary layer, , can be defined at the height above the surface where the flow velocity on the boundary layer approaches the external or edge flow velocity, .

The boundary layer typically begins as a laminar layer, at least over a smooth surface with a low-disturbance freestream, where the fluid motion is orderly and mixing normal to the wall is weak. As the laminar boundary layer thickens downstream, it may become unstable to small disturbances. The location of transition, which usually occurs over a finite distance rather than at a single point, depends primarily on the Reynolds number, but it is also affected by surface roughness, pressure gradient, wall temperature, freestream turbulence, and acoustic or mechanical disturbances. After transition, the boundary layer becomes turbulent and is characterized by strong mixing across much of its thickness. Even then, a very thin viscous sublayer, often historically called the laminar sublayer, remains adjacent to the wall, where viscous stresses dominate, and the mean velocity varies approximately linearly with distance from the wall.

Notice that while turbulent boundary layers have a greater thickness than laminar boundary layers, it should be remembered that the velocity profiles of laminar and turbulent boundary layers are also different. While boundary layers are of two primary types, laminar and turbulent, the third type is transitional. A transitional boundary layer is not fully developed because it is neither laminar nor turbulent. However, it typically persists in this form only over a relatively short zone or downstream distance.

Distance-Based Reynolds Number

The developing nature of the boundary layer can be characterized in terms of a local Reynolds number, which is measured in terms of the distance from the point where the flow initially develops on the surface, i.e.,

(5)  

For example, as previously discussed, is measured from the leading edge of a flat plate to some downstream distance. The boundary layer often develops downstream from a stagnation point on a body, as illustrated in Figure 8. Generally, the downstream distance differs between the upper and lower surfaces and depends on the surface geometry.

Transition

Experiments on smooth external flows over surfaces have shown that the boundary layer generally transitions to turbulence at a Reynolds number based on of about . The reason is that natural flow disturbances develop even on perfectly smooth surfaces, including those with mildly favorable pressure gradients, leading to a transition to a turbulent boundary layer. In this case, the value of the “critical” Reynolds number for transition to be initiated is denoted by , and the corresponding value of is denoted by , i.e.,

(6)  

and so

(7)  

Many experiments on airfoils and wings have examined the transition from laminar to turbulent boundary-layer flow. In all cases, the transition region has been found to correlate strongly with the Reynolds number as a function of downstream distance. For a smooth NACA 0012 airfoil, the critical Reynolds number for laminar-turbulent transition is of the order of 10 at moderate to high angles of attack. However, the critical Reynolds number will depend somewhat on the airfoil’s shape. On airfoils, transition often occurs downstream of the suction peak, where the boundary layer encounters an adverse pressure gradient, but its location depends strongly on Reynolds number, pressure distribution, angle of attack, surface roughness, freestream turbulence, and the disturbance environment.^[1] Laminar boundary layers are sensitive to adverse pressure gradients because of their low velocities near the wall.

Because turbulent mixing during transition progresses gradually, the transition from a fully laminar to a fully turbulent boundary layer occurs over a finite distance. It is not a fixed point, per se. Although this distance is usually relatively small, it is still finite and depends on the Reynolds number, pressure gradient, surface condition, and disturbance environment. In some cases, transition occurs through a laminar separation bubble. Such bubbles are especially important at low-to-moderate chord Reynolds numbers, often below about 10. However, short bubbles may also occur at higher Reynolds numbers under suitable pressure-gradient and disturbance conditions.

Other Properties of Boundary Layers

Besides the velocity profile or in the boundary layer, several additional parameters are used to quantify its overall aerodynamic effects. These parameters describe how the boundary layer alters the effective shape of a body, how it changes the flow momentum near the surface, and how close the flow may be to separation. For this purpose, engineers commonly use three integral measures of the boundary layer, namely:

1. The displacement thickness, .
2. The momentum thickness, or .
3. The shape factor, .

Displacement Thickness

The displacement thickness of a boundary layer is given the symbol and is defined by

(8)  

the idea of which is illustrated in Figure 9.

Notice that for conservation of mass in the boundary layer per unit depth, then

(9)  

where it will be remembered that is the value of when . On rearrangement, this gives

(10)  

and, finally,

(11)  

Therefore, the displacement thickness of a boundary layer can be interpreted as equivalent to the height or thickness corresponding to the reduction in mass flow associated with the boundary layer profile, which is, in effect, a mass flow deficit relative to the freestream flow.

Momentum Thickness

The corresponding momentum thickness of the boundary layer, which is given the symbol or , is defined by

(12)  

In this case, the momentum thickness represents the loss of streamwise momentum flux caused by the lower velocity within the boundary layer. Per unit span, the momentum-flux deficit is

(13)  

Therefore,

(14)  

or, equivalently,

(15)  

This momentum thickness parameter also has physical significance, as it quantifies the reduction in momentum relative to the freestream because of boundary-layer development, i.e., the momentum deficit in the flow. It also provides information about the momentum distribution within the boundary layer, i.e., whether it is distributed closer to or further from the wall.

Shape Factor

Finally, the shape factor, given the symbol , is defined in terms of the displacement thickness and momentum thickness by the equation

(16)  

The shape factor is often used as an indicator of the state of boundary-layer flows and to evaluate their tendency to remain attached or to separate from surfaces. Notice that because within the boundary layer, it follows that

(17)  

then the value of is always greater than unity. Experiments have shown that varies with flow state. For example, for laminar flows, ; for fully turbulent flows, . As a boundary layer approaches separation, increases, but there is no single universal value of that defines separation. Laminar separation is often associated with values on the order of 3.5-4, whereas turbulent separation may occur at lower values, depending on the pressure-gradient history. Therefore, the shape factor is a useful diagnostic parameter for assessing the state of a boundary layer and its susceptibility to separation.

Pressure Gradients & Flow Separation

A pressure gradient in any given flow will significantly affect the development of a boundary layer. As illustrated in Figure 10, three types of pressure gradients can develop over a surface: neutral (zero), favorable, and adverse.

1. A pressure gradient where the pressure decreases with increasing downstream distance, say , i.e., the pressure gradient is . This gradient is termed a favorable pressure gradient because it accelerates the flow, reduces the rate of boundary-layer thickening, increases resistance to separation, and generally helps keep the flow attached.

2. A neutral or “zero” pressure gradient where , i.e., no pressure changes with downstream distance.

3. A pressure gradient where the pressure increases with increasing downstream distance, i.e., . This gradient is called an adverse pressure gradient because it has adverse effects on the boundary layer; i.e., the pressure forces act to retard its development, thereby slowing it. If the pressure gradient is sufficiently adverse, it will eventually cause the boundary layer to slow to zero near the surface and separate from it.

Figure 11 illustrates the development of boundary-layer flow over a convex hump. The external flow velocity (in this case identified by the edge velocity ) increases over the front of the hump and reduces again over the back. As the flow velocity increases, the static pressure decreases (i.e., according to the Bernoulli equation), and as the velocity decreases, the pressure increases again. Consequently, the corresponding pressure gradients are favorable over the front ( is negative, decreasing with distance) and adverse over the back ( is positive, increasing with distance). The favorable pressure gradient over the front half of the flow keeps the boundary layer attached. Still, the adverse pressure gradient over the back half of the flow slows the boundary-layer flow, especially near the surface, causing it to separate. Notice that when flow separation occurs, the flow velocity and static pressure do not immediately recover to the freestream values.

The consequence of the preceding observations is that the flow no longer easily follows the body’s contour when the gradient , i.e., what happens when the pressure gradient is adverse and tries to slow down the flow? There comes a point where the velocity gradient at the wall becomes zero, so the corresponding wall shear stress vanishes. Downstream of this point, the near-wall flow may reverse, and a recirculating region develops. It is said that the flow separates when the shear stress vanishes; the point at which this occurs is called the flow separation point. The formal criterion for the onset of flow separation is that the wall shear stress vanishes, i.e.,

(18)  

or, equivalently,

(19)  

This means the flow separates from the surface when the wall shear stress is zero. Although this criterion is strictly valid only for steady flow, it provides a quantifiable condition for predicting or identifying the onset of flow separation in a boundary layer.

It can be inferred that turbulent boundary layers are generally less susceptible to flow separation than laminar boundary layers, with all external influences equal. This outcome arises because turbulent mixing transports higher-momentum fluid from the outer part of the boundary layer toward the wall, producing a fuller velocity profile and a larger near-wall velocity gradient. Therefore, a turbulent boundary layer can usually withstand a stronger adverse pressure gradient before the wall shear stress falls to zero.

Flow Separation on an Airfoil

Boundary-layer separation from a body, such as an airfoil section, can have significant consequences, including a substantial increase in drag and a substantial loss of lift. This outcome is due to the rear part of an airfoil creating an adverse pressure gradient, as shown in Figure 12, which becomes increasingly adverse as the angle of attack increases. Although pressure is a scalar quantity, in this presentation it is plotted as a vector oriented perpendicular to the airfoil surface, with lower-than-ambient static-pressure areas pointing outward and higher-than-ambient static-pressure areas pointing inward.

Along the leading edge of the airfoil, a favorable pressure gradient promotes the normal downstream development of the boundary layer. Indeed, the boundary layer may remain laminar during this period, although this depends on surface roughness. Beyond the minimum pressure point, an adverse pressure gradient develops over the airfoil. This gradient decelerates the boundary-layer flow and may promote transition, laminar separation, or both, depending on the Reynolds number and disturbance environment. Subsequently, the boundary layer decelerates downstream as it progresses in the adverse pressure gradient.

The boundary layer can withstand this gradient at low angles of attack, reaching the airfoil’s trailing edge or separating just before that point, as shown in Figure 13. However, as the angle of attack increases, the separated region generally grows and may extend progressively farther forward over the airfoil. Eventually, the resulting separation causes a substantial loss of lift, and under these conditions, the airfoil is considered stalled.

The turbulence produced in the separated-flow region and wake is also a source of unsteady aerodynamic loads and wing buffeting. Indeed, stalling an airplane wing during flight induces unsteady aerodynamic loads and buffeting, which are transmitted to the airframe and warn the pilot of an impending wing stall.

Surface Roughness

The development of a boundary layer is profoundly affected by pressure gradients and surface roughness. For example, the transition from a laminar to a turbulent boundary layer is often prematurely triggered by surface roughness, as shown in Figure 14. This enhances mixing in the lower layers of the boundary layer, leading to the quicker development of turbulence. However, a laminar boundary layer is so thin that even a small amount of roughness can initiate transition. Leading-edge roughness can trip the boundary layer earlier and may suppress or shorten laminar separation bubbles, depending on the roughness height, location, Reynolds number, and pressure-gradient distribution.

As this turbulence develops near the surface, it mixes upward through the boundary layer’s higher layers. Consequently, it quickly progresses through the entire boundary layer from top to bottom, except for the immediate vicinity of the wall. Surface roughness can occur for various reasons, including the “in-service” use of an aircraft. For example, average abrasion at the leading edge of a wing increases the drag of that wing. However, in some cases, such as with golf balls, surface roughness can have beneficial effects; the transition to a turbulent boundary layer delays flow separation and reduces the pressure drag on the ball.

Summary of Boundary Layer Characteristics

At this stage, it is worthwhile to pause and reflect on the essential features of the boundary layer outlined so far. These concepts help clarify the fundamental behavior of viscous flows near solid surfaces and form the basis for understanding drag, transition, and separation. A deeper understanding of boundary layers, which often becomes highly mathematical, will build directly on these ideas; therefore, it is essential to grasp the basics before proceeding. The key ideas are:

1. The flow-velocity profile and other boundary-layer characteristics, such as thickness, influence the shear stresses in the fluid and the surface shear, also known as skin friction. Hence, the boundary layer affects the drag of the surface (or body) over which it develops.
2. Laminar and turbulent boundary layers exhibit distinct behaviors. Boundary layers may begin as laminar but often transition to turbulence at some distance downstream, depending on the Reynolds number, surface conditions, and freestream disturbances. Turbulent boundary layers have fuller velocity profiles and higher momentum near the wall, which makes them more resistant to separation but also increases the surface shear stress and skin-friction drag.
3. As fluid flows along a surface, the boundary layer thickens in the streamwise direction. This growth reflects the progressive diffusion of viscous effects into the outer flow and is a key factor in determining the flow development and pressure distribution over a body.
4. The characteristics of the boundary layer depend on the Reynolds number. At higher Reynolds numbers, where inertial effects in the flow are much stronger than viscous effects, boundary layers tend to be relatively thin and robust, and they stay attached to surfaces over longer downstream distances. However, at lower Reynolds numbers, the boundary layers are thicker, slower, and more prone to separation.
5. The characteristics of the boundary layer are affected by the pressure gradients over the surface on which it develops. For example, if the surface is curved rather than flat, the flow velocity and static pressure will vary along it. If the pressure gradient is sufficiently adverse, the boundary layer may detach from the surface, a process called flow separation.
6. Boundary layers become susceptible to separation when they encounter sufficiently strong adverse pressure gradients, particularly after they have thickened and lost momentum near the wall. The onset of flow separation has deleterious effects on the aerodynamics of airfoils and wings, including a loss of lift and an increase in drag. This behavior, when observed on an airfoil or wing, is called a stall. Other effects, such as surface roughness and flow heating, may also significantly influence the development of the boundary layer.

Boundary Layer Equations

The “boundary layer equations” are a subset of the full Navier-Stokes equations, which, in principle, are more straightforward (but not easy) to solve, partly because they are cast into a two-dimensional form using order-of-magnitude approximations. Such approximations require careful justification by comparing the resulting approximate solutions with the complete solutions (if available), flow measurements, or both. Experience with the boundary-layer equations has demonstrated their validity for problems subject to the assumptions and limitations under which they were derived. For example, using the boundary-layer equations to solve for viscous flow over an airfoil at low to moderate angles of attack (below stall) is well established.

Many practical problems can also be solved by using a hybrid or coupled method. This method combines essentially inviscid methods (e.g., potential-flow methods, such as panel methods for the outer flow) with a boundary-layer approach for the inner, near-surface flow. These are sometimes referred to as inviscid/viscous interaction methods. In this approach, the boundary-layer solution is coupled to the inviscid solution via a specific boundary condition. This is typically expressed as the boundary-layer displacement thickness, which provides a “virtual” airfoil shape over which the inviscid solution can be obtained. Such approaches can provide reasonable predictions of the resulting flow at a lower cost than a complete Navier-Stokes solution using computational fluid dynamics (CFD).

Momentum Equations

Different approaches can be used to derive the boundary-layer equations. The most intuitive approach is to start with the incompressible form of the momentum equations and to apply them to a problem (such as the flow over an airfoil at lower angles of attack) where the boundary layer development can be assumed to be very thin (e.g., what would be obtained with a flow at higher chord Reynolds numbers). The streamline curvature is low, i.e., the streamlines are not bent by much, so the slope of the streamlines, , remains small.

The incompressible form of the momentum (Navier-Stokes) equations can be written in a rather elegant and compact form as

(20)  

where is the vector Laplacian of the velocity field. The equations are also often written in the form

(21)  

Expanding the substantial derivative on the left-hand side gives

(22)  

The meaning of the terms in these equations should be appreciated, i.e.,

(23)  

The terms refer to normal compressive stresses, such as pressure, and viscous shear. Note that the normal compressive stresses in terms of the bulk viscosity do not appear in the incompressible form of the equations.

These equations can also be expanded in terms of their scalar components, giving

(24)  

Two-Dimensional, Steady Flow

For a two-dimensional flow (assumed in the – plane), there are just two scalar components, i.e.,

(25)  

Expanding out the first and last set of terms in each case gives

(26)  

and assuming steady flow, then

(27)  

Simplifications

Consider a boundary layer whose thickness, , is much smaller than the characteristic streamwise length, , i.e.,

(28)  

For an airfoil, the characteristic length may be taken as the chord, so that . The streamwise velocity is of the same order as the external velocity , whereas the normal velocity is much smaller. From the continuity equation,

(29)  

the characteristic velocity scales are

(30)  

and

(31)  

so that . Because changes in the streamwise direction occur over a distance of order , while changes normal to the wall occur over a distance of order ,

(32)  

and

(33)  

The two convective terms in the streamwise momentum equation are then of the same order, i.e.,

(34)  

Therefore, both convective terms must be retained.

The viscous diffusion terms scale as

(35)  

and

(36)  

Because , it follows that

(37)  

Therefore, viscous diffusion in the streamwise direction can be neglected relative to viscous diffusion normal to the wall.

Under the same thin-boundary-layer assumptions, the normal momentum equation shows that the pressure variation across the boundary layer is negligible to first order. Therefore,

(38)  

and, in the classical incompressible boundary-layer approximation, this result is written as

(39)  

which implies that . Therefore, the pressure within the boundary layer at a given streamwise station is equal to the pressure at the edge of the boundary layer. The external flow imposes the streamwise pressure gradient and is not determined independently by the boundary-layer solution.

These approximations reduce the steady, two-dimensional streamwise momentum equation to

(40)  

together with

(41)  

and the continuity equation

(42)  

Thin-Layer Boundary Layer Equations

The external inviscid flow imposes the streamwise pressure gradient. Applying the streamwise Euler equation at the edge of the boundary layer gives

(43)  

Substituting this expression into the reduced streamwise momentum equation gives the standard form of the steady, incompressible boundary-layer equations, i.e.,

(44)  

These equations must be solved together with the continuity equation for incompressible flow, i.e.,

(45)  

If time dependence is included, the pressure gradient imposed by the external flow is

(46)  

and the boundary-layer equations take the unsteady form

(47)  

These equations collectively form the basis of classical laminar boundary-layer theory, which describes the dominant balance of forces in the thin shear layer adjacent to a surface. They apply directly to steady or unsteady incompressible laminar boundary layers before separation. For turbulent boundary layers, the same thin-layer scaling is used, but the equations are usually written in Reynolds-averaged form with additional Reynolds-stress terms to account for turbulent momentum transport. The boundary-layer approximations cease to be valid in strongly separated flows, such as those on stalled airfoils or in the wakes behind bluff bodies.

Although the boundary-layer equations are a significantly reduced form of the Navier-Stokes equations, they remain challenging to solve because they are nonlinear, with the nonlinearity arising from the convective acceleration terms. They are typically solved numerically, although exact similarity solutions are possible in some cases. In all cases, appropriate initial and boundary conditions must be specified.

Laminar Boundary Layer – Blasius Solution

A classic solution in this latter category is the Blasius solution^[2] for the laminar boundary layer development over a flat plate in a zero pressure gradient, which has become a classic exemplar in fluid dynamics and an essential check case for CFD and measurement methods. As shown in Figure 15, the flow away from the plate has a uniform velocity of . The downstream distance is measured in terms of , and the plate’s leading edge is at = 0. The boundary layer develops from the leading edge of the ideal flat plate, where the boundary-layer thickness is taken to be zero in the mathematical model, and thickens as it moves downstream. This problem had been studied experimentally in the laboratory by the time Blasius developed his theoretical solution, providing additional physical insight that informed his theory.

Boundary Layer Equations in Zero Pressure Gradient

The boundary layer equations in a zero-pressure gradient are

(48)  

so these equations become

(49)  

Notice that the first of the set (Eq. 49) is in terms of the kinematic viscosity , i.e., . These latter equations are sometimes known as Prandtl’s equations because Prandtl was the first to cast the more general momentum equations into this simplified form.

Stream Function & Boundary Conditions

To solve the Prandtl equations for the flow field in the boundary layer, a set of boundary conditions is needed, which, in this case, are: 1. The no-slip and no-penetration conditions at the surface, i.e., and at . 2. The conditions at infinity (a uniform flow in this case), i.e., as . Using the stream function, , is a useful way to reduce the number of variables, and Blasius employed this approach in his analysis of the boundary layer problem. His goal was to reduce the governing partial differential equations (which, in this case, are expressed in terms of and ) to a single ordinary differential equation for a single transformed variable. Remember that in terms of the stream function, , the velocity components are

(50)  

Notice that the continuity equation is automatically satisfied, i.e.,

(51)  

Equation 49 becomes

(52)  

which is now a new governing equation. It is important to note that the physical boundary conditions must be rewritten in terms of the stream function. At the wall, the no-slip condition requires , and the no-penetration condition may be satisfied by setting at . Far from the wall, the boundary condition becomes as .

Similarity Solution

Blasius’s next step was to express the velocity solution in the boundary layer as a self-similar function of downstream distance, thereby enabling consistent scaling of the boundary-layer profile. Blasius examined the boundary-layer profile measurements at different downstream distances, , when plotted as versus , where is the free-stream velocity and is the boundary-layer thickness. The results proved to be self-similar, which means that the velocity profile at any value of is a scaled version of the velocity profile at any other distance . When the velocity profiles were plotted as versus the scaled coordinate , they collapsed into a single profile, indicating that they exhibited self-similarity when expressed in terms of these quantities.

George Stokes had already shown that for a boundary layer, the flow velocity at a distance from the surface would depend on , the kinematic viscosity , and the distance from the leading edge of the plate , and the boundary layer thickness, , grew according to

(53)  

Dividing both sides by , then

(54)  

so the developing thickness of the boundary layer becomes a function of the Reynolds number based on downstream distance, i.e.,

(55)  

Solution of the Velocity Profile

The primary goal remains, however, which is to determine the actual theoretical shape of the boundary layer profile that will satisfy the governing boundary layer equation to find as a function of , i.e.,

(56)  

where is the desired functional relationship. From this point, Blasius defined a transformed variable, , as

(57)  

A dimensionless stream function is then introduced through

(58)  

In terms of the transformed variable, then

(59)  

and

(60)  

The governing equation, i.e., Eq. 52, now becomes

(61)  

or in shorthand notation

(62)  

The boundary conditions in terms of the transformed variables become and at , corresponding to the no-penetration and no-slip conditions, respectively, while as , corresponding to the condition in the external flow, as summarized in the table below.

Transformation of boundary conditions from physical to similarity coordinates.

Physical coordinate	Similarity coordinates
= 0 at = 0	= 0 at = 0
= 0 at = 0	= 0 at = 0
as	= 1 as

Notice that the transformed governing equation, i.e., Eq. 62, is still a third-order nonlinear differential equation, but it is now an ordinary differential equation (ODE), which is easier to solve. Therefore, Blasius’s approach reduced a system of two partial differential equations to a single ODE. However, solving Eq. 62 for the function still cannot be done in closed form and requires a numerical method for which several approaches are possible. Blasius did this using a Taylor series expansion for which the coefficients of the series were numerically derived using recursive relationships, and much of his dissertation work is about finding this numerical solution.

However, other numerical methods can also be used to solve the equation. For example, the third-order ODE in Eq. 62 can be converted into a system of first-order ODEs by letting

(63)  

Therefore, the original Blasius equation in Eq. 62 now becomes

(64)  

so, the system of three ODEs to be solved is

(65)  

Also, the new initial conditions are

(66)  

This system of ODEs can now be integrated numerically (e.g., using a Runge-Kutta method) from to a sufficiently large value of , such as , which would be well outside of the boundary layer thickness. Paul Blasius would surely have appreciated the benefits of MATLAB and the ode45 function. The required value is approximately . The value must be selected so that the far-field boundary condition is satisfied. The code below uses the known approximate value rather than determining it iteratively.

MATLAB code to solve the Blasius equation

Show code/hide code.

% Define the Blasius equation as an equivalent ODE system
% Boundary conditions
eta_max = 10;
initial_guess = 0.332; % Initial guess for h(0)
% Define the range of eta
eta_range = [0 eta_max];
% Initial conditions [f(0), f'(0), f”(0)]
initial_conditions = [0, 0, initial_guess];
% Solve the ODE using ode45
[eta, y] = ode45(@blasiusODE, eta_range, initial_conditions);
% Extract the solution
f = y(:,1);
g = y(:,2);
% Plot the results
figure;
plot(eta, g, ‘-b’, ‘DisplayName’, ‘f”(\eta)’);
hold on;
plot(eta, f, ‘-r’, ‘DisplayName’, ‘f(\eta)’);
xlabel(‘\eta’);
ylabel(‘f, f”’);
legend;
title(‘Solution to the Blasius Equation’);
grid on;
hold off;
function dydeta = blasiusODE(eta, y)
f = y(1);
g = y(2);
h = y(3);
dydeta = [g; h; -0.5 * f * h];
end

The table below presents selected numerical values of the similarity function and its derivatives as functions of the similarity coordinate , continuing until approaches 1, i.e., the edge of the boundary layer. The value of the second derivative, , is needed to calculate skin friction on the surface of the plate from the action of the boundary layer.

The Blasius similarity function and its derivatives.

0.0	0.0	0.0	0.332
0.5	0.042	0.166	0.331
1.0	0.166	0.330	0.323
1.5	0.370	0.487	0.303
2.0	0.650	0.630	0.267
2.5	0.996	0.751	0.217
3.0	1.397	0.846	0.161
3.5	1.838	0.913	0.108
4.0	2.306	0.956	0.064
4.5	2.790	0.980	0.034
5.0	3.283	0.992	0.016
5.5	3.781	0.997	0.007
6.0	4.280	0.999	0.002
		1.0	0.0

Finally, notice that the velocity profile of the laminar boundary layer is just

(67)  

which is the result that Blasius set out to obtain from the outset. This velocity profile is plotted in Figure 16. Measurements show that the Blasius velocity profile solution is in excellent agreement with those obtained using particle image velocimetry (PIV). Quod erat demonstrandum.

Other Laminar Boundary Layer Characteristics

From this result, additional information about the laminar boundary layer, such as its thickness and wall shear stress, can be obtained. The boundary layer thickness, , is defined as the value of at which the boundary-layer velocity approaches 99% of the freestream value. From the Blasius solution, this condition is obtained for = 5 so that

(68)  

or in terms of , where is the Reynolds number based on “,” which is the distance from the plate’s leading edge, then

(69)  

The displacement thickness, , is obtained by integrating over the velocity profile, i.e.,

(70)  

giving

(71)  

where 1.721 is found from the numerical solution. Therefore, the quantity can be written as

(72)  

Also, the momentum thickness, or , is

(73)  

or

(74)  

It will be apparent that the values of and can be expressed in terms of the boundary layer thickness, i.e.,

(75)  

and

(76)  

The shear stress on the surface, or wall stress, is

(77)  

and from the tabulated numerical values given previously, = 0.332. Therefore,

(78)  

It is often preferable to use non-dimensional forms of aerodynamic quantities. When dealing with boundary layers, a local non-dimensional shear stress coefficient or skin friction coefficient is used, which is defined in terms of the freestream dynamic pressure as

(79)  

The value of will be a local quantity because it will vary from point to point across the surface, so the value of will also vary.

The local skin-friction coefficient is integrated to calculate the total drag on an exposed surface from the boundary layer. The drag on a plate of finite length would be given by integrating the value of the local skin friction along the length of both sides of the plate, i.e.,

(80)  

For an airfoil section of chord , then

(81)  

This latter result is often used as a benchmark because it represents the absolute minimum drag on a thin airfoil if the flow developed along its length in a fully laminar manner. Therefore, the goal is to design for laminar flow over as much of the surface as practical. However, maintaining extensive laminar flow at practical flight Reynolds numbers requires favorable pressure gradients, low surface roughness, low waviness, and a low-disturbance environment; otherwise, the boundary layer will transition to turbulence.

Falkner-Skan Solutions

Extensions of Blasius’s similarity approach have been developed for laminar boundary layers subjected to pressure gradients. A well-known generalization is the Falkner-Skan solution, which applies when the external velocity profile varies with distance along the surface according to

(82)  

where is a parameter that is related to the pressure gradient in the external flow. For this velocity distribution, an appropriate similarity transformation can again reduce the boundary-layer equations to a single ODE. The resulting Falkner-Skan equation^[3] is

(83)  

where is a dimensionless stream function and is the Falkner-Skan similarity variable describing the scaled distance normal to the surface. The Blasius similarity variable is recovered as the special case for a zero pressure gradient. The velocity within the boundary layer is obtained from

(84)  

so that represents the nondimensional velocity profile. The associated boundary conditions are

(85)  

which correspond to the no-penetration and no-slip conditions at the wall and the matching of the boundary-layer velocity to the external flow. When = 0, the equation reduces to the Blasius flat-plate solution corresponding to a zero pressure gradient. Positive values of correspond to favorable pressure gradients, whereas negative values correspond to adverse pressure gradients.

Figure 17 shows that the Falkner-Skan solutions illustrate how pressure gradients modify the velocity distribution within a laminar boundary layer. The velocity profiles for several values of the Falkner-Skan parameter are plotted as the nondimensional velocity versus . When the pressure gradient is favorable , the boundary layer becomes thinner, and the velocity profile becomes fuller, reaching the freestream velocity more rapidly. When the pressure gradient is adverse , the profiles become flatter, the velocity gradient at the wall decreases, and the boundary layer thickens. As the adverse pressure gradient increases, the wall shear stress approaches zero and separation occurs when -0.09.

These laminar similarity solutions are mathematically elegant and provide useful insight into how pressure gradients influence boundary-layer behavior. However, their direct practical use is limited because the boundary layers on most flight vehicles transition to turbulence over much of their surfaces. In general, because of their lower near-wall momentum, laminar boundary layers are more susceptible to separation under adverse pressure gradients than turbulent boundary layers. If a separated laminar shear layer transitions to turbulence and reattaches, then a laminar separation bubble may form.

Turbulent Boundary Layers

The principles governing the development of a turbulent boundary layer, which is characterized by strong mixing between fluid layers and a more uniform velocity profile, have already been discussed. Recall that a turbulent boundary layer is generally thicker than a laminar one because of the increased mixing in the flow. However, modeling the detailed structure of a turbulent boundary layer is much more complex mathematically, and there are no theoretical solutions comparable to the Blasius solution for laminar flow.

Velocity Profiles

Turbulent boundary layers have been studied extensively through experiments, but developing a simple analytical expression for the velocity profile remains challenging. Nevertheless, several empirical approximations are available. One of the simplest and best-known approximations is the power-law profile, which can be written as

(86)  

where the exponent depends on the Reynolds number. Experimental measurements show that increases slowly with Reynolds number. For many practical applications, taking = 7 provides a reasonable representation of a turbulent boundary-layer profile, commonly referred to as the one-seventh power law, as illustrated in Figure 18.

A limitation of the power-law profile is that it cannot accurately calculate the wall shear stress because it predicts an infinite velocity gradient at the wall ( = 0).
In reality, a thin region adjacent to the wall known as the viscous sublayer exists in which the mean velocity varies approximately linearly with distance from the wall. This behavior is usually expressed in wall variables as

(87)  

where

(88)  

Therefore, in the viscous sublayer,

(89)  

which is consistent with Newton’s law of viscosity at the wall. For a turbulent boundary layer developing along a flat plate, the boundary-layer thickness may be approximated by

(90)  

This relationship assumes that the flow is turbulent from the leading edge and that the velocity profiles are self-similar along the -direction. Neither assumption is strictly valid for most practical flows, so the formula should be used with appropriate caution. Further details on turbulent boundary layers are given in the chapter on turbulence later in this eBook.

Effects of Pressure Gradients

The influence of pressure gradients on turbulent boundary layers can be understood from the Reynolds-averaged boundary-layer momentum equation, i.e.,

(91)  

where represents the Reynolds shear stress arising from turbulent velocity fluctuations. This term accounts for turbulent momentum transport and generally dominates over much of the turbulent boundary layer. However, the Reynolds shear stress approaches zero at the wall, where the viscous shear stress determines the wall stress.

As in laminar boundary layers, the pressure gradient is determined by the external inviscid flow and is related to the variation of the external velocity by

(92)  

When , the pressure decreases along the surface, and the flow experiences a favorable pressure gradient. In this case, the boundary layer tends to become thinner, and the wall shear stress increases. When , the pressure increases along the surface and the flow experiences an adverse pressure gradient, causing the boundary layer to thicken and the velocity gradient at the wall to decrease.

In turbulent boundary layers, the effects of pressure gradients are also reflected in the structure of the mean velocity profile. Turbulent mixing continually transports high-momentum fluid from the outer flow toward the wall, allowing the near-wall region to retain relatively high momentum even in the presence of an adverse pressure gradient. As the adverse pressure gradient strengthens, the outer portion of the velocity profile becomes increasingly distorted, and the velocity deficit relative to the external flow grows.

Boundary-layer separation occurs when the velocity gradient at the wall becomes zero, i.e.,

(93)  

Because turbulent mixing replenishes the near-wall momentum, turbulent boundary layers can withstand substantially stronger adverse pressure gradients before separating than laminar boundary layers.

Clauser Parameter

The strength of the pressure gradient acting on a turbulent boundary layer is often characterized using the Clauser pressure-gradient parameter, defined as

(94)  

where is the displacement thickness and is the wall shear stress. This parameter provides a convenient nondimensional measure of the pressure gradient’s influence on the boundary layer.

As shown in Figure 19, when , the boundary layer experiences little or no pressure-gradient effect and behaves similarly to the zero-pressure-gradient turbulent boundary layer over a flat plate. As increases, the adverse pressure gradient becomes stronger, the boundary layer thickens, and the velocity profile becomes increasingly distorted relative to the external flow. Large values of indicate strong adverse pressure gradients and conditions approaching boundary-layer separation.

Boundary Layer Growth in Wind Tunnel Test Sections

Understanding and estimating boundary-layer parameters, such as the displacement thickness, are essential in aerodynamic design and testing. For example, in the design of wind-tunnel test sections, a critical consideration is the growth of boundary layers along the walls, as illustrated in Figure 20. As the boundary layers thicken downstream, they effectively displace the outer flow toward the center of the test section. This displacement reduces the effective flow area and increases the core-flow velocity, which in turn produces an unintended pressure gradient along the test section.

To maintain a uniform velocity and nearly zero pressure gradient in the test section, the geometric cross-sectional area must be adjusted to compensate for the growth of the wall boundary layers. In practice, this correction is implemented by slightly diverging the walls or by incorporating corner fillets that gradually taper along their length. These geometric adjustments offset the increasing displacement thickness of the boundary layers, keeping the effective aerodynamic flow area nearly constant along the test section.

By accounting for boundary-layer growth in this way, wind tunnels can provide a nearly uniform freestream velocity along the test section, which is essential for obtaining accurate and repeatable aerodynamic measurements. The key parameter for quantifying the boundary layer’s effect on the flow is the displacement thickness, , i.e.,

(95)  

where is the streamwise velocity profile, and is the velocity at the edge of the boundary layer. Recall that can be interpreted as the equivalent distance by which the external flow streamlines are displaced because of the slower-moving fluid in the boundary layer near the wall.

Consider first a constant-area rectangular test section of width and height . If boundary layers grow on all four walls, then the effective flow area available to the approximately inviscid core flow is reduced to

(96)  

where represents the displacement thickness on the side walls and represents the displacement thickness on the floor and ceiling. Therefore, by continuity, assuming incompressible flow and constant mass flow,

(97)  

where

(98)  

and is the core-flow velocity at the beginning of the test section. Therefore,

(99)  

Thus, in a constant-area test section, boundary-layer growth reduces the effective flow area, causing the core velocity to increase downstream.

To maintain a nearly uniform core velocity along the test section, the geometric area must be increased with downstream distance so that the effective area remains approximately constant. If the corrected test-section dimensions are written as and , then the design condition is

(100)  

or, equivalently,

(101)  

Expanding the effective-area expression gives

(102)  

If the displacement thicknesses are small compared to the test-section dimensions, i.e.,

(103)  

then the product term is second order and can be neglected. Therefore,

(104)  

For small wall divergence angles, one may use the uncorrected dimensions and in the correction terms to obtain the approximate required increase in geometric area,

(105)  

so that

(106)  

The values of and must be estimated from the boundary-layer development on the corresponding walls. For fully turbulent boundary layers in an approximately zero pressure gradient, the displacement thickness may be estimated using

(107)  

where

(108)  

is the Reynolds number based on the distance along the test section.

In addition to area correction, attention must be paid to the corner regions, where boundary layers from adjacent walls interact to form corner vortices that can distort the velocity field. To mitigate this, corner fillets are often installed, either as quarter-cylindrical or optimized contours, to reduce corner vortex formation and streamline the flow. Tapered fillets are usually the best solution because they mitigate corner flow effects and control the geometric area. In this way, the four sides of the test section can be made parallel, and the tapered fillets can be designed to meet the requirements.

von Kármán Momentum Integral Equation

The momentum integral method is used when exact flow solutions are not possible, e.g., non-zero pressure gradient flows, turbulent boundary layers, flows with surface curvature or blowing/suction, and flows over arbitrary bodies where only the outer (e.g., potential) flow is known, e.g., in terms of the velocity and the static pressure . The momentum integral equation is often used in zonal methods, in which an outer potential-flow solution is coupled to a boundary-layer solution, e.g., via the displacement thickness, and the two solutions are iterated until compatibility is achieved.

The incompressible, two-dimensional thin-layer boundary layer equations are

(109)  

and

(110)  

Assume the pressure gradient is imposed by the inviscid outer flow and extends into the boundary layer. The basic principles of pressure balance and integrated changes in momentum within the boundary layer are shown in Figure 21.

To derive the von Kármán momentum integral equation, the boundary-layer momentum balance must be written in terms of finite deficit quantities. This point is important because outside the boundary layer, so integrals of or alone over are not finite. The relevant finite integral measures are the displacement thickness and momentum thickness, defined by

(111)  

and

(112)  

or, equivalently,

(113)  

The upper limit is conventionally written as because the integrands are boundary-layer deficit quantities that vanish outside the boundary layer. Therefore, the integrals remain finite even though the boundary layer velocity approaches asymptotically.

For a steady, incompressible, two-dimensional boundary layer, integration of the momentum equation across the boundary layer gives the integral momentum balance

(114)  

where

(115)  

is the shape factor. Because , the same equation may also be written as

(116)  

Finally, multiplying both sides by gives the von Kármán momentum integral equation, i.e.,

(117)  

This is the famous von Kármán momentum integral equation for two-dimensional boundary layer flow, which provides a global, integral expression of momentum conservation within the boundary layer. Each term has a direct physical interpretation, i.e.,

• The term represents the growth of the boundary layer’s momentum deficit in the streamwise direction.
• The term accounts for the influence of a pressure gradient imposed by the external flow.
• The right-hand side represents the loss of momentum because of viscous shear at the wall.

This equation avoids solving the full boundary-layer equations and instead uses integrated properties, making it highly useful for practical engineering analysis. It serves as the foundation for several approximate boundary-layer methods, including Thwaites’ method for laminar boundary layers, various empirical turbulent boundary-layer models, and predictions of the onset of flow separation. Unlike the Blasius solution, which is restricted to zero-pressure-gradient flows, the von Kármán equation accommodates arbitrary external pressure gradients, making it broadly applicable to real aerodynamic flows.

Visualizing the Boundary Layer

Boundary layers are typically very thin regions near a surface, making them difficult to visualize using measurement techniques. Nevertheless, standard flow-visualization methods in wind tunnels can reveal signatures and specific boundary-layer properties. Flow visualization can be divided into surface and off-surface visualization. Such methods include, but are not limited to, smoke or other tracer particles, tufts, surface oil flows, liquid crystals, sublimating chemicals, pressure-sensitive paints, shadowgraph, schlieren, etc.

Figure 22 illustrates the effects of the two boundary-layer states on an airfoil at a low angle of attack and a relatively low Reynolds number. In this case, the flow was visualized in a wind tunnel using a surface oil-flow technique, in which a brush was used to apply a light oil to the surface, and the surface was illuminated with ultraviolet (UV-A) light. The shear stress from the boundary layer on the oil mixture creates a pattern, with regions of lower skin friction leaving behind more significant oil accumulations.

Under these conditions, the forward 60% of the airfoil can be interpreted as having a laminar boundary layer; the low surface shear stress here allows oil to accumulate, particularly near the mid-chord, where the boundary layer approaches the transition point. Downstream of the transition, the boundary layer is fully turbulent, with higher shear stresses removing more of the oil from the surface. Notice that the transition region is short where the flow is separated, indicative of a laminar separation bubble. However, it must be remembered that what is observed on the surface does not always indicate what is happening in the external flow.

Skin Friction Drag on an Airfoil

Estimating the skin-friction drag on bodies with turbulent boundary layers is an essential problem in aerospace engineering. While the situation is generally complex for a complete aircraft or other flight vehicles because of the intricate surfaces and the highly three-dimensional boundary layers that develop, the process can be illustrated for a two-dimensional airfoil; see Figure 23. An airfoil is reasonably thin and flat, so at low angles of attack, a flat plate can approximate the shape and drag of an airfoil.

The net shear stress drag on the plate with the laminar flow can be found by integrating the local shear along the plate’s length or the chord . For the Blasius result, then the sectional drag coefficient is

(118)  

where the factor “2” is needed because the plate has upper and lower surfaces; this latter result is plotted for reference in Figure 23.

For the fully turbulent boundary layer development on a flat plate, the skin friction coefficient on one side of the plate is found to conform to an empirical relationship given by

(119)  

This latter result is a common empirical estimate associated with the one-seventh-power-law approximation for a smooth, fully turbulent flat-plate boundary layer. If the plate were to have a fully turbulent boundary layer over its upper and lower surfaces, then by integration (as for the laminar case), it is found that

(120)  

where is the chord-based Reynolds number. The validity of the latter expression is generally limited to a range between and . Below , a smooth flat-plate boundary layer in a low-disturbance flow may remain predominantly laminar. However, transition can occur earlier because of surface roughness, freestream turbulence, vibration, or adverse pressure gradients, and laminar separation bubbles may form.

The results shown in the graph above suggest that the turbulent flat-plate solution is a good approximation to the viscous (shear) drag on many airfoils over a fairly wide range of practical Reynolds numbers found during flight on aircraft, i.e., for above . In this case, it will be sufficient to assume that

(121)  

where is the reference chord Reynolds number for which a reference value of drag is known for a given airfoil section; not all airfoils will have known (measured) drag coefficients at all Reynolds numbers, but this approach allows the drag coefficient for other Reynolds numbers to be estimated with good confidence.

The empirical drag equation suggested by McCroskey for practical use is

(122)  

This expression offers an improved correlation at higher chord Reynolds numbers, which are typical of those found on larger aircraft and at higher flight speeds. The laminar boundary layer separates more readily at very low Reynolds numbers, below . The drag coefficients of typical airfoils are larger than those predicted by either laminar or turbulent boundary-layer theory.

Airfoil behavior in this Reynolds-number regime is essential for many classes of small-scale air vehicles, e.g., UAVs. In this case, Figure 23 shows that changing the scaling coefficient from -0.2 to -0.4 in Eq. 121 gives a good approximation (line fit) based on the results shown over a lower range of . However, the drag of airfoils in this regime is less predictable, particularly at angles of attack other than shallow angles, because the boundary layers are thicker, often with laminar separation bubbles, and the onset of flow separation occurs more readily.

Laminar Separation Bubbles

The term “laminar separation bubble” has been mentioned previously. Figure 24 shows a schematic of a laminar separation bubble, or LSB; an LSB can occur on airfoil sections over a wide range of chord Reynolds numbers, . However, LSBs are usually longer and more pronounced at lower Reynolds numbers, especially below about 10. Laminar separation bubbles occur when a laminar boundary layer separates under an adverse pressure gradient, often downstream of a suction peak in the surface pressure distribution on the suction surface of the airfoil. The separation location depends on airfoil shape, angle of attack, Reynolds number, surface condition, and the imposed pressure-gradient history. After laminar separation occurs, the flow temporarily leaves the airfoil surface, undergoes transition, and then reattaches as a turbulent boundary layer. This reattachment occurs because a turbulent boundary layer can withstand a given adverse pressure gradient more readily than a laminar boundary layer. The result is a region of recirculating separated flow, usually relatively small at higher chord Reynolds numbers, over which the surface pressure is nearly constant. Accordingly, an LSB may also appear as a slight pressure plateau in the chordwise pressure distribution, often near the airfoil’s leading edge.

At lower Reynolds numbers, a long LSB may form. Using surface oil-flow visualization, a previously discussed method, an LSB is revealed by a local accumulation of oil, as shown in the photograph in Figure 25. At a chord Reynolds number in this case of about 10⁴, the LSB is relatively long as a fraction of the chord. After laminar separation, the wall shear stress is very low and may even become negative locally within the separated region, allowing the oil to accumulate in the bubble. When the flow reattaches farther downstream, the higher shear stress convects much of the remaining oil toward the trailing edge, leaving a residual accumulation within the LSB. As the angle of attack increases, the LSB often moves toward the leading edge and shortens, although with a further increase in angle of attack it may burst, leading to more extensive separation and possibly stall. At still lower Reynolds numbers, the bubble can lengthen considerably.

The formation of an LSB begins with the external velocity distribution imposed on the boundary layer by the pressure distribution over the airfoil. For incompressible flow, the pressure gradient follows from

(123)  

so an adverse pressure gradient, , corresponds to . The development of the boundary layer can be described approximately by the momentum-integral equation

(124)  

where is the momentum thickness, is the boundary-layer shape factor, and is the local skin-friction coefficient. As the laminar boundary layer decelerates, its velocity profile becomes less full, increases, and the wall shear stress decreases. Laminar separation occurs at when

(125)  

or, equivalently, when .

Downstream of , the separated laminar boundary layer forms a free shear layer above a region of reverse flow. This shear layer is unstable, so small disturbances grow rapidly and cause the flow to transition from laminar to turbulent. After transition, turbulent mixing transfers higher-momentum fluid toward the surface. If this momentum transfer is sufficient to overcome the adverse pressure gradient, the mean wall shear stress becomes positive, and the flow reattaches at . The closed region between and is the laminar separation bubble, with length

(126)  

The pressure within most of the bubble is nominally constant because the separated shear layer largely isolates the surface from the external pressure gradient. However, the displacement effect of the bubble modifies the effective airfoil shape, the external velocity distribution , and the surface pressure distribution, so the inviscid external flow and viscous boundary layer must generally be solved interactively. If transition occurs too late for turbulent reattachment, the bubble bursts and develops into an extensive region of separated flow.

Boundary Layer Flow Control

Boundary-layer flow control refers to methods used to influence the development, transition, or separation of a boundary layer to improve aerodynamic performance. The objectives may include maintaining laminar flow to reduce skin-friction drag, delaying separation to increase lift or reduce pressure drag, and controlling unsteady separated flow. These objectives are related but are not identical because a turbulent boundary layer produces more skin-friction drag than a laminar boundary layer but is also more resistant to separation.

One approach is natural laminar flow, in which the shape of the airfoil or body is designed to produce a favorable or only mildly adverse pressure gradient over a substantial downstream distance. This pressure distribution suppresses the growth of disturbances and allows the boundary layer to remain laminar without active control. However, natural laminar flow is sensitive to surface roughness, waviness, insect contamination, manufacturing tolerances, and freestream disturbances. It is also difficult to maintain over swept wings because crossflow instabilities can promote transition.

A second approach is laminar-flow control, in which suction is applied through slots or a porous surface to remove low-momentum fluid from the boundary layer. Suction reduces the boundary-layer thickness, increases the velocity gradient near the wall, and suppresses the growth of disturbances that would otherwise cause transition. In principle, suction can maintain laminar flow farther downstream than would be possible through airfoil shaping alone. A combination of favorable pressure-gradient design and suction is known as hybrid laminar-flow control.

NASA tested this concept on a modified F-16XL using a porous glove installed over part of the wing, as shown in Figure 26. The glove contained millions of very small holes connected to an internal suction system. Air was drawn through the surface to remove low-momentum fluid from the boundary layer and extend the region of laminar flow over the highly swept wing. The experiment demonstrated the aerodynamic potential of suction-based laminar-flow control, although practical implementation requires pumps, ducts, filters, porous surfaces, and contamination protection.

Boundary-layer control may also be used to delay or suppress flow separation. Suction can remove the low-momentum fluid that accumulates near the wall, while blowing or fluidic jets can add momentum to the boundary layer. Passive devices such as vortex generators produce streamwise vortices that transport higher-momentum fluid from the outer flow toward the surface. These devices generally cause a small drag penalty when the flow is attached but can delay separation and reduce the much larger drag increase associated with separated flow.

The overall value of a boundary-layer control system depends on the net aerodynamic and energy benefits. Any reduction in drag must be weighed against the mass, power consumption, complexity, surface contamination, reliability, and maintenance requirements of the system. Consequently, successful boundary-layer control requires an integrated assessment of aerodynamics, structures, propulsion, and aircraft operations.

Thermal Boundary Layers

A thermal boundary layer is a concept in fluid dynamics and heat transfer that describes the region near a surface where the fluid temperature changes rapidly from the surface (wall) temperature to the temperature of the external bulk fluid. Heat is transferred when a fluid flows over a wall at a temperature different from that of the fluid. This heat-transfer process creates a region adjacent to the wall, known as the thermal boundary layer, where the temperature gradient is pronounced.

Heat Transfer

Assume that a flow with freestream temperature starts to flow over a wall with a higher temperature , i.e., . At the hotter wall, the flow velocity is low and approaches the no-slip condition at the surface, as shown in Figure 27. Because there is a temperature difference between the wall and the flow, heat is first transferred across the wall/fluid interface and into the fluid by thermal conduction. Thermal energy is conducted away from the wall into the fluid, which is then carried downstream by the flow, producing a thermal boundary layer in which the temperature gradually approaches the external-flow temperature. The thermal thickness, , is usually defined as the distance from the wall where the temperature has recovered to within 1% of the external-flow temperature, i.e.,

(127)  

for the case . The rate at which the thermal boundary layer grows differs from the viscous or velocity boundary layer. If the freestream temperature is higher than the surface temperature, i.e., , heat is transferred from the fluid to the surface. At the wall-fluid interface, this transfer occurs by thermal conduction, while convection transports thermal energy through the moving fluid.

The rate of heat conductive transfer from the wall per unit area, also known as heat flux (which has units of energy per unit time per unit area), depends on the temperature gradient at the wall, often called Fourier’s law. In its one-dimensional form, it is expressed as

(128)  

where is known as the thermal conductivity of the fluid (which has units of energy per unit time per length per unit temperature). The minus sign indicates that heat flows in the opposite direction of the temperature gradient from higher to lower, i.e., in accordance with the second law of thermodynamics. The thermal conductivity, , depends on the fluid type and temperature and is a measured quantity obtained from heat-transfer experiments.

Nusselt Number

The heat transfer per unit area of the wall can also be expressed in terms of Newton’s law of cooling, i.e.,

(129)  

where, in this case, is the convective heat transfer coefficient (expressed in units of energy per unit time per unit area per unit temperature), a value that can also be determined from experiments. Because the conductive heat transfer is transferred as convection into the thermal boundary layer, equating Eqs. 128 and 129 gives

(130)  

This latter relationship in Eq. 130 can be non-dimensionalized by using

(131)  

and

(132)  

where is some characteristic length. Therefore, after some additional steps, it can be shown that

(133)  

which is a dimensionless temperature gradient, the dimensionless grouping being known as the Nusselt number, i.e,

(134)  

Recall that is the thermal conductivity of the fluid, and is the convective heat-transfer coefficient, so the physical interpretation of the Nusselt number becomes apparent. In boundary-layer heat-transfer correlations, the reference length for the Nusselt number is usually a streamwise length scale, such as the distance from the leading edge of a wing, , for a local Nusselt number or the chord length, , for an average Nusselt number, i.e.,

(135)  

Prandtl Number

In the study of thermal boundary layers, the Prandtl number, , is often used, which is a dimensionless similarity parameter defined as the ratio of viscous or “momentum” diffusivity to thermal diffusivity, i.e.

(136)  

where is the thermal diffusivity, a measure of how rapidly temperature disturbances diffuse through the fluid, and is the specific heat at constant pressure. The product is called the volumetric heat capacity, i.e., the fluid’s ability to absorb and store thermal energy. The momentum diffusivity, normally called kinematic viscosity, is a measure of the fluid’s resistance to shear relative to its density.

Therefore, the Prandtl number is a relative measure of momentum diffusion to thermal diffusion. For , thermal diffusivity is more important, and for , momentum diffusivity is more important. Notice that the Prandtl number has no length scale, and its value depends solely on the temperature-driven values of the fluid parameters. Its value and other properties, such as density, viscosity, and thermal conductivity, are usually listed.

For common gases such as air, ; for air near ambient conditions, . For a laminar boundary layer, thermal diffusivity is greater than momentum diffusivity, so . The molecular Prandtl number depends on the fluid properties and temperature, not on whether the flow is laminar or turbulent. In turbulent flows, an analogous turbulent Prandtl number is often introduced to model the relative turbulent transport of momentum and heat; for air, it is commonly taken to be of order 0.9, although its value depends on the flow and turbulence model. For many common liquids, ; for example, water near ambient conditions has , while viscous oils may have .

The thickness of the thermal boundary layer, , depends on factors such as the flow velocity, specific fluid properties, and the surface temperature. Generally, a thicker thermal boundary layer results in slower heat transfer, while a thinner boundary layer allows for quicker heat transfer between the wall and the fluid. Turbulence enhances heat transfer by more effectively mixing the fluid near the surface, bringing more low-temperature fluid into contact with the hot surface, thereby increasing the thermal gradient and the overall heat transfer rate. Therefore, for laminar boundary layers over a flat plate, the relative thicknesses of the velocity and thermal boundary layers scale approximately as

(137)  

for fluids with moderate Prandtl numbers.

Importance of Thermal Boundary Layers

Understanding thermal boundary layers is critical for high-speed flight vehicles to manage aerodynamic heating, select structural materials, and ensure structural integrity. Supersonic and hypersonic flight generate intense aerodynamic heating from compressibility and wall shear. Thermal boundary layers form around the aircraft’s surfaces as a result of this heating, affecting the temperature distribution and the thermal stresses experienced by the aircraft structure. For example, the leading edges of Concorde reached temperatures over 130^oC (see Figure 28), which was enough to begin altering the material properties of conventional aluminum; to this end, a special high-temperature “Hiduminium” alloy had to be used. Metals are also very good at conducting heat, so the entire airframe reached relatively high temperatures, causing it to expand thermally by up to 0.5 ft (0.15 m) from nose to tail. To this end, many parts of the internal structure used pin-jointed and/or sliding elements to alleviate the stresses.

During re-entry into Earth’s atmosphere or when entering the atmospheres of other planets, spacecraft experience intense aerodynamic heating, primarily from compression and shock-layer heating of the gas ahead of the vehicle, followed by convective and radiative heat transfer to the surface. Understanding thermal boundary layers is crucial for designing effective heat shields that dissipate heat efficiently and prevent the spacecraft from burning up during re-entry. Some heat shields are made of ablative material that degrades in a controlled manner, absorbing energy and carrying heat away with the ablated material. Spacecraft may employ alternative thermal protection systems, such as silica tiles, ceramics, and other coatings, capable of withstanding high temperatures. These systems often rely on the principles governing thermal boundary layers to dissipate heat efficiently and protect the spacecraft’s structure and components.

The surface temperatures of the Space Shuttle, as shown in Figure 29, reached 1,300 K during re-entry at Mach 14. During the early stages of re-entry, the atmosphere is thin, and the Reynolds numbers are relatively low, so portions of the boundary layer may remain laminar. A laminar boundary layer generally produces lower convective heating than a turbulent boundary layer, whereas transition to turbulence increases near-wall mixing and can substantially increase the heat-transfer rate to the surface. In this flight, a boundary-layer trip was applied to the lower surface of the port wing, downstream of the leading edge, to generate and study turbulence; the subsequent higher-temperature zone (in red) indicates that disturbed turbulent flow significantly increases surface temperatures. Therefore, maintaining laminar flow as long as possible during re-entry is crucial for reducing thermal loads and ensuring the effectiveness of the thermal protection system. This goal is achieved through aerodynamic design, surface characteristics, and suitable re-entry trajectories.

Summary & Closure

Understanding boundary layers is crucial for optimizing the aerodynamic performance of both lifting and non-lifting bodies. Laminar boundary layers, characterized by smooth, orderly flow, produce lower skin-friction drag but are more susceptible to flow separation, particularly under adverse pressure gradients. While laminar boundary layers are easier to understand, the reality of most flows about flight vehicles is that their boundary layers are fully turbulent. Turbulent boundary layers have higher skin friction but are also more resistant to flow separation, thereby helping the flow remain attached and delaying separation and stall, especially on airfoils and wings.

While the study of turbulent boundary layers is a technical field in itself, understanding their essential characteristics and behavior is critical to comprehending the aerodynamic characteristics of airfoils, wings, other shapes, and flight vehicles as a whole. Techniques such as wind tunnel testing, advanced computational fluid dynamics (CFD), and advances in turbulence modeling can help predict boundary-layer behavior. Consequently, understanding and controlling boundary layers is fundamental to achieving improved aerodynamic efficiency, fuel savings, and overall flight vehicle performance.

Understanding thermal boundary layers is also essential for the design of high-speed aircraft and re-entry spacecraft. Keeping the boundary layer laminar for as long as possible reduces turbulent mixing near the surface and usually lowers the convective heat-transfer coefficient, thereby reducing heat transfer to the surface and helping to limit structural temperatures. When the boundary layer transitions to turbulence, flow mixing increases, enhancing heat transfer to the surface and leading to higher surface temperatures. Predicting the thermal boundary layer and the regions of laminar versus turbulent flow is essential when selecting structural and protective materials capable of withstanding the temperatures encountered during high-speed flight or re-entry.