Boundary Layers

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n22

23 Boundary Layers

Introduction

In many flows over surfaces, including internal and external surfaces, viscosity effects are confined to a thin flow region near the surface, known as the boundary layer. The boundary layer is a fundamental concept in aerodynamics and fluid dynamics, in general, because it substantially influences a body’s overall aerodynamic characteristics, including its lift and drag. For example, the concept of the boundary layer as it develops over the surface of an airfoil is illustrated in the figure below. However, it should be remembered that boundary layers are very thin and much thinner in reality than the zones shown in this figure.

The concept of a boundary layer, in this case as it develops over the surface of an airfoil, which is indicated by the orange zone near the surface. A boundary layer is very thin, even in comparison to the thickness of an airfoil.

One effect produced by the boundary layer development over a body is to create shear stresses on its surface, the cumulative manifestation of shear being a type of drag called skin friction drag. Skin friction is a significant source of overall drag on an aircraft, particularly in high-speed flight, where it can account for about half of the total drag. Another effect is that boundary layers can detach or separate from surfaces under certain conditions, which is a phenomenon called flow separation. The onset of boundary layer flow separation from a surface generally has a deleterious effect on the aerodynamics with a loss of lift and the creation of drag. In many cases, the aerodynamic characteristics of bodies can only be explained in relation to the underlying boundary layer behavior.

Learning Objectives

Understand the basic concept of a boundary layer and appreciate its physical significance as it can affect aerodynamic flows over airfoils, wings, and other body shapes.
Distinguish between the primary characteristics of both laminar and turbulent boundary layers.
Know how to calculate the shear stress on a surface under the influence of a boundary layer for particular types of boundary layer velocity profiles and be able to estimate skin friction drag.
Understand why and under what conditions boundary layers may detach or separate from surfaces and what subsequent effects may be produced.

History

Historically, the concept of the boundary layer was introduced by Ludwig Prandtl. His work on the subject originated from experiments made with his students in Germany at the beginning of the 20th century. Prandtl showed that for fluids having relatively small values of viscosity that flow over surfaces or bodies, the effects of the internal stresses in the fluid are appreciable only in a very thin region near the fluid boundaries, i.e., in a region, at the surface he called the boundary layer.

Understanding boundary layers and their effects on flow developments was a quantum step forward in growing knowledge in the field of fluid dynamics. Before this, in the 17th century, Isaac Newton discovered that fluids could be deformed. Because of the relative motion between their layers, then shear stresses are created in the fluid, resulting in what is now known as Newton’s law of viscosity. Newton also noted no relative motion between the fluid and a solid surface (the “wall”), which is called the no-slip condition.

As shown in the figure below, the flow velocity in a boundary layer rises quickly from zero at the surface (no-slip condition) and then smoothly and asymptotically approaches the external flow velocity, $V_e$ , as the distance increases away from the surface. However, it should be appreciated that a boundary layer is very thin; the vertical extent or thickness of the boundary layer is given the symbol $\delta$ . The boundary layer is said to have a shape or profile, i.e., the velocity in the boundary layer parallel to the surface, $u$ , varies as a function of the distance from the surface $y$ , i.e., $u = u(y)$ . The functional form of the boundary layer profile depends on many factors, including the nature of the surface and the effects produced by the external flow, e.g., the pressure gradients.

Concept of a boundary layer as it forms adjacent to a solid surface or “wall,” where the flow velocity increases smoothly and asymptotically with distance away from the wall.

Different types of boundary layers will have different velocity profiles, but, as will be discussed, they are usually characterized into either laminar or turbulent boundary layer profiles, each having similar but also different flow characteristics. In general, what goes on inside the boundary layer is very important in aerodynamics and fluid dynamics. One is that it affects the shear produced on a body’s surface over which the boundary layer develops and, therefore, the magnitude of the net skin friction drag produced on the body. The properties of the boundary layer also affect whether the flow stays attached to the body’s surface or may separate from it.

Laminar & Turbulent Boundary Layers

As detailed in the figure below, in a laminar boundary layer, the fluid flows in an orderly manner, with smooth layers of fluid free of any mixing between successive layers; such layers are called laminae, from which the “laminar” name is derived. The fluid layers in a turbulent boundary layer become mixed, so the flow velocities away from the immediate region at the wall tend to be more uniform. Because of the rotational motion of the fluid and the shear stresses that result from that, vortical eddies and turbulence are also formed, creating significant mixing in the flow. Turbulent boundary layers are inevitably thicker than laminar boundary layers because the effects of flow mixing extend further away from the wall.

Basic differences in the characteristics between a laminar boundary layer and a turbulent boundary layer.

A more useful quantitative comparison of the profile shapes of a typical laminar and turbulent boundary layer is shown in the following figure. This presentation is obtained by normalizing the velocity profile by the boundary layer thickness $\delta$ and the outer or edge velocity of the external flow, $V_e$ , i.e., by plotting $u/V_e$ as a function of $y/\delta$ . Note that in the case of a turbulent boundary layer, $u$ would represent the mean or time-averaged flow velocity, not the instantaneous velocity.

Normalized velocity profiles for laminar and turbulent boundary layers, which are distinctly different.

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

(1) $\begin{equation*} \left( \frac{u}{V_e} \right) = f \left( \frac{y}{\delta} \right) \end{equation*}$

so that the value of $u/V_e$ becomes unity at the edge of the boundary layer, i.e., when $y/\delta = 1$ .

Because the velocity in the boundary layer smoothly and asymptotically approaches the flow velocity of the external flow, the value of $\delta$ must be defined rather carefully to avoid ambiguity; by convention, $\delta$ is defined as the value of $y$ for which 99% of the external flow velocity is recovered, i.e., when $u = 0.99 V_e$ . Notice that the velocity gradients, i.e., $d u / d y$ , in a laminar boundary layer are relatively shallow throughout.

Shear Stresses & Skin Friction

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

(2) $\begin{equation*} \tau = \mu \left( \frac{du}{dy} \right) \end{equation*}$

where $\du / dy$ is the rate at which the flow velocity increases (in the $y$ direction) and is equivalent to a strain rate, as shown in the figure below. A characteristic of fluids is that they must be continuously deformed (strained) to produce shear, unlike solid materials, where a strain will always maintain a stress.

The shear stress is proportional to the velocity gradient, this being equivalent to a strain-rate in the fluid. Notice the “gradient” shown in this case is $d y / d u$ or ( $d u / d y)^{-1}$ , so the shallower this gradient the higher the value of $d u / \, d y$ .

Notice that the shear stress is proportional to $d y / d u$ , so the shallower this gradient, the higher the value of $d u / \, d y$ . Because of the mixing in a turbulent boundary layer, its velocity profile is more uniform 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, $\delta$ , than a laminar boundary layer that forms under the same flow conditions.

Equation 2 is a very general expression, and it will apply to any point in the flow. However, it is restricted to fluids that behave such that they follow a linear stress/strain relationship, i.e., a so-called Newtonian fluid where the slope of the stress/strain-rate relationship is linear, and so the value of $\mu$ is constant.

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 $y \rightarrow 0$ , because this value determines the skin friction drag produced by the boundary layer on the surface. The stress in the fluid at the wall, which is in the upstream direction, has a reaction shear on the wall in the downstream direction. At the wall, the shear stress is

(3) $\begin{equation*} \tau = \tau_w = \mu \left(\frac{d u}{d y}\right) \mbox{ as $y \rightarrow 0$} \end{equation*}$

or more usually just as

(4) $\begin{equation*} \tau = \mu \left( \frac{d u}{d y} \right)_{y \rightarrow 0} \end{equation*}$

As implied by the shape of the boundary layer velocity profiles, as shown in the figure below, the magnitude of the wall shear stress, $\tau_w$ , 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.

The shear stress on a surface produced by a flowing laminar boundary layer (a) is much lower than that produced by a turbulent boundary layer (b).

This latter result is significant because the development of the boundary layer is the source of the shear stress drag or skin friction drag on a flight vehicle. Therefore, it would be desirable to generate low drag by having a laminar boundary layer over as much of the vehicle as possible. However, in reality, it is found that laminar boundary layers do not exist for very long in terms of the downstream distance over which they develop, and so they become turbulent.

Nevertheless, calculating the shear stress caused by the boundary layer is a prerequisite to calculating the drag on any body shape exposed to a flow. As might be expected, drag prediction is difficult for an entire aircraft or other flight vehicles because it has many complex shapes and three-dimensional flows. The boundary layers also take on more complicated three-dimensional forms in this case.

Development of a Boundary Layer

It should be appreciated that a boundary layer is not a static phenomenon, and it is said to “grow'” as it develops with downstream distance, i.e., the thickness and other properties of a boundary layer will change continuously as it develops downstream over any given surface. Furthermore, the development of a boundary layer may also be affected by other things, 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 flow of uniform velocity. Because the surface is flat without curvature, no pressure gradients will be produced over the surface.

As shown in the figure below, the thickness of the boundary layer is zero at the plate’s leading edge. Its thickness grows progressively and asymptotically, meaning that its growth rate is initially significant, and then the growth rate slows down as the boundary layer develops further downstream. Be aware that the vertical scale used in this figure has been exaggerated for clarity, and boundary layers are physically very thin. As previously mentioned, the thickness of the boundary layer, $\delta$ , can be defined at the height above the surface where the flow velocity on the boundary layer approaches the external or edge flow velocity, $V_e$ .

Schematic showing the development of a boundary layer over a flat plate. The flow initially develops as a laminar boundary layer, but transitions to a turbulent boundary layer. The location of transition, which happens over a finite distance or zone, depends on the downstream distance based on Reynolds number.

The boundary layer begins in laminar form, at least over smooth surfaces, in which the fluid moves in smooth layers or laminae. As the laminar boundary layer increases in thickness as it develops downstream, then the layers of fluid naturally tend to mix. Finally, the boundary layer transforms itself into a well-mixed turbulent boundary layer. Even when the boundary layer becomes turbulent, there is still a very thin layer next to the wall with laminar flow, called the laminar sublayer.

Notice that while turbulent boundary layers have a greater thickness than laminar boundary layers, 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 of boundary layer can be considered to be the transitional type. A transitional boundary layer is not a fully developed type of boundary layer because it is not fully laminar or fully turbulent. However, usually, it exists in this form only for a relatively short zone or downstream distance.

Example #1 – Finding Surface Shear from the Velocity Profile

The velocity profiles in a laminar boundary layer have been determined to take on one of the three following characteristic non-dimensional forms, which are plotted in the figure below:

$\mbox{Profile A:} \quad \frac{u}{V_e} = \frac{y}{\delta}$

$\mbox{Profile B:} \quad \frac{u}{V_e} = \sin \left( \frac{\pi}{2} \frac{y}{\delta} \right)$

$\mbox{Profile C:} \quad \frac{u}{V_e} = 2 \left( \frac{y}{\delta}\right) - \left(\frac{y}{\delta} \right)^2$

where $V_e$ is the flow velocity at the edge of the boundary layer, $y$ is the distance from the surface, and $\delta$ is the boundary layer thickness. Find the relative shear stress on the wall is to be determined for given values of $\delta$ and $V_e$ .

Notice that all three profiles are of the “laminar” type with relatively low or shallow velocity gradients throughout. Notice that this is also one-dimensional flow problem with velocities that only vary in the $y$ direction, so the shear stress on the wall in each case is given by evaluating

$\tau_w = \mu \left( \frac{du}{dy} \right)_{y/\delta\rightarrow 0} = \mu \left(\frac{V_e}{\delta} \right) \left( \frac{d (u/V_e)}{d (y/\delta)} \right)$

Therefore, for Profile A:

$\frac{d (u/V_e)}{d (y/\delta)} = 1$

and for Profile B:

$\frac{d (u/V_e)}{d (y/\delta)} = \frac{\pi}{2} \cos \left( \frac{\pi}{2} \frac{y}{\delta} \right) = \frac{\pi}{2} \mbox{ as $y/\delta\rightarrow 0$}$

and finally for Profile C:

$\frac{d (u/V_e)}{d (y/\delta)} = 2 \left(1 - (y/\delta) \right) = 2 \mbox{ as $y/\delta\rightarrow 0$}$

On the basis of the same value of $\delta$ and also the same external flow velocity $V_e$ then the grouping

$\mu \left(\frac{V_e}{\delta} \right) = \mbox{constant}$

is the same so it becomes clear that Profile A has the lowest relative shear stress and Profile C has the highest, which is readily confirmed by reinspection of the results plotted in the figure above.

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 $x$ from the point where the flow initially develops, i.e.,

(5) $\begin{equation*} Re_ x = \frac{\varrho_{\infty} V_{\infty} x}{\mu_{\infty} } = \frac{V_{\infty} x}{\nu_{\infty}} \end{equation*}$

For example, in the case of a flat plate, as previously discussed, then $x$ is measured from the plate’s leading edge to some downstream distance.

Experiments on smooth surfaces have shown that at Reynolds numbers based on $Re_x$ above about $5 \times 10^{5}$ the boundary layer generally always becomes turbulent. The reason is that natural flow disturbances develop even over perfectly smooth surfaces and cause a transition from a laminar to a turbulent boundary layer. In this case, the value of the “critical” Reynolds number for transition to occur is denoted by $Re_{\rm cr}$ and the corresponding value of $x$ is denoted by $x_{\rm cr}$ , i.e.,

(6) $\begin{equation*} Re_ {\rm cr} = \frac{\varrho_{\infty} V_{\infty} x_{\rm cr}}{\mu_{\infty} } \end{equation*}$

Many experiments have examined the transition process from laminar to turbulent flows. In all cases, the point or region of transition correlates strongly to the Reynolds number based on downstream distance. Because the turbulent mixing occurs progressively, the transition from a fully laminar boundary layer flow to one fully turbulent is a process that will occur over some distance. Although this distance is usually relatively small, it is still finite.

Pressure Gradients & Flow Separation

A pressure gradient in any given flow will have significant effects on the development of a boundary layer. There are three types of pressure gradients that can develop over a surface, as illustrated in the figure below.

The concept of a pressure gradient. If the static pressure increases with downstream distance then this is referred to as an adverse pressure gradient because it slows up the development of a boundary layer. If pressure decreases with downstream distance then it is called a favorable pressure gradient because it tends to encourage the continued development of a boundary layer.

1. A pressure gradient where the pressure $p$ decreases with increasing downstream distance, say $x$ , i.e., the pressure gradient is $dp/dx < 0$ . This gradient is called a favorable pressure gradient because it has favorable effects on the boundary layer developments, i.e., it tends to encourage the downstream development of the boundary layer and prevent it from separating from the surface.

2. A neutral or “zero” pressure gradient where $dp/dx = 0$ , i.e., there is no change in the pressure with downstream distance.

3. A pressure gradient where the pressure $p$ increases with increasing downstream distance, i.e., $dp/dx > 0$ . This gradient is called an adverse pressure gradient because it has adverse effects on the development of the boundary layer, i.e., the pressure forces act to retard the development of the boundary layer and so tend to slow it up. If the pressure gradient is sufficiently adverse, it will eventually cause the boundary layer to slow down to zero near the surface and separate away from the surface.

The figure below shows an example of a boundary layer flow over a convex hump. The external flow velocity (in this case identified by the edge velocity $V_e$ ) increases over the front of the hump and reduces again over the back. As the flow velocity increases, its static pressure decreases (i.e., from the Bernoulli equation), and as the velocity decreases again, the pressure increases. The consequence of this is that the corresponding pressure gradients are favorable over the front ( $dp/dx$ is negative, decreasing with distance) and adverse over the back ( $d p/d x$ is positive, so increasing with distance). The favorable pressure gradient over the front half encourages the boundary layer to stay attached, but the adverse pressure gradient over the back half slows the boundary layer and causes it to separate.

Flow over a hump. The flow velocity increases to a maximum as it flows over the crest, with the corresponding pressure reaching a minimum.

On the one hand, it can be seen that the favorable pressure gradient tends to maintain or even increase the velocity in the boundary layer close to the wall. However, on the other hand, it can be seen that the other side’s adverse pressure gradient slows the flow velocity at the wall and will eventually cause the boundary layer to separate away from the surface.

The consequence of the preceding observations is that the flow no longer easily follows the body’s contour when the gradient $(d u/d y)_w \rightarrow 0$ , i.e., it is adverse and tries to slow down the flow. There comes the point that the boundary layer profile develops a point of inflection, so the corresponding shear stress becomes zero. The flow reverses, and a region of recirculating flow develops. Now it is said that the flow has separated, and the point where the shear stress is zero is called the flow separation point. Notice that when flow separation occurs, the flow velocity and static pressure do not immediately recover to the free-stream values.

The formal criterion for the onset of flow separation is that

(7) $\begin{equation*} \tau_w = \left(\frac{d u}{d y} \right)_w = 0 \end{equation*}$

which means that when the wall shear becomes zero then the flow will separate from the surface. While this criterion is strictly valid only for a steady flow, it represents a quantifiable condition to predict or otherwise identify the onset of flow separation in a boundary layer.

It is possible to deduce that turbulent boundary layers are generally less susceptible to flow separation than a laminar boundary layer, all external influences being equal. This outcome is because a turbulent boundary layer has a higher flow velocity near the wall than a laminar boundary layer, so it takes a longer downstream distance for the flow to approach the conditions at the wall that are likely to produce flow separation.

Summary of Boundary Layer Characteristics

The preceding observations about the boundary layer can now be summarized:

1. The flow velocity profile and other characteristics of the boundary layer, such as its thickness, affect the shear stresses in the fluid and the surface shear or skin friction. Hence, the boundary layer affects the drag of the surface (or body) over which it develops.

2. The boundary layer’s characteristics 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 to the effects that may cause separation. However, at lower Reynolds numbers, the boundary layers are slower and thicker and so more easily prone to separation.

3. The characteristics of the boundary layer are affected by the pressure gradients that exist over the surface in which the boundary layer develops, e.g., if the surface is curved rather than flat, then the flow velocities will change along the surface, and so will the static pressures. If the pressure gradient is sufficiently unfavorable or adverse, the boundary layer may detach or separate and leave the surface, which is called flow separation.

4. Most boundary layers will eventually become susceptible to separation from the surfaces over which they develop, especially after they develop over longer downstream distances. The onset of flow separation has deleterious effects on airfoil and wing aerodynamics, including a loss of lift and an increase in drag. Other effects may also be significant to the development of the boundary layer, e.g., surface roughness, flow heating, etc.

Flow Separation on a Lifting Surface

Boundary layer separation from a body, such as an airfoil section, as shown in the schematic below, can have significant consequences, including a large increase in drag and a significant loss of lift. This outcome is because the rear part of an airfoil creates an adverse pressure gradient, which becomes increasingly adverse with an increasing angle of attack. At low angles of attack, the boundary layer can withstand this gradient and either reaches the airfoil’s trailing edge or separates just before that point. However, as the angle of attack increases, the more severe adverse pressure gradients cause the flow to separate at a shorter downstream distance, so the flow separation point moves forward. Eventually, flow separation occurs near the leading edge, and under these conditions, the airfoil is said to be stalled.

Flow about an airfoil at increasing angles of attack (angles are exaggerated for clarity) showing the progressive onset of flow separation and the development of separated flow containing recirculation of the flow and turbulence.

The turbulence produced in the separated flow region and wake is also a source of unsteady aerodynamic loads and buffeting on the wing. Indeed, a characteristic of stalling the wing of an aircraft during flight is the creation of unsteady aerodynamic loads and buffeting that are transmitted to the airframe and so give the pilot warning of an impending wing stall.

Laminar Boundary Layer: Blasius Solution

The development of a laminar boundary layer in a zero pressure gradient flow over a smooth flat surface can be calculated precisely. This result was first obtained by Paul Blasius, a student of Ludwig Prandtl, and his theoretical result has become a classic reference for understanding laminar boundary layer flows. The problem studied by Blasius is summarized in the figure below. Although Blasius provided an “exact” theoretical result for the laminar boundary layer, certain assumptions that were justified from experiments were used when formally establishing the theory. This input included the “self-similarity” of the laminar boundary layer profiles, i.e., the velocity profiles are the same when non-dimensionalized by their thickness.

As already discussed, it is often preferred to use non-dimensional forms of quantities in aerodynamics, e.g., lift coefficient, drag coefficient, and the like. When dealing with boundary layers, a local non-dimensional shear stress coefficient or skin friction coefficient $c_f$ is used, which is defined in terms of the free-stream dynamic pressure as

(8) $\begin{equation*} c_{f} = \frac{\tau_w}{q_{\infty}} = \frac{\tau_w}{\frac{1}{2} \varrho V_{\infty}^2} \end{equation*}$

The value of $\tau_w$ will be a local quantity because it will vary from point to point across the surface, so the value of $c_f$ will also vary. The local skin friction coefficient is used with integration to calculate total drag on an exposed surface from the action of the boundary layer.

In Blasius’s solution, the local skin friction coefficient, $c_{f}$ , on one side of a flat plate is found to be

(9) $\begin{equation*} c_{f} = \frac{\tau_w}{\frac{1}{2} \varrho V_{\infty}^2} = 0.664 Re_{x}^{-0.5} \end{equation*}$

where $Re_{x}$ is the Reynolds number based on “x” which is the distance from the leading edge of the plate. Other results that come from the Blasius solution include the boundary layer thickness, which is

(10) $\begin{equation*} \delta = 5 \sqrt{ \frac{ \nu x}{V_{\infty}}} = 5 Re_x^{1/2} \end{equation*}$

and the corresponding displacement thickness $\delta^*$ and momentum thickness $\theta$ are

(11) $\begin{equation*} \delta^* = \int_0^\delta \left( 1 - \frac{u}{U_{\infty}} \right) dy = 1.721 \sqrt{ \frac{ \nu x}{U_{\infty}}} \approx 0.344 \delta \end{equation*}$

and

(12) $\begin{equation*} \theta = \int_0^\delta \frac{u}{U_{\infty}} \left( 1 - \frac{u}{U_{\infty}} \right) dy = 0.664 \sqrt{ \frac{ \nu x}{U_{\infty}}} \approx 0.128 \delta \end{equation*}$

However, while an interesting result, laminar boundary layers do not exist in practice for values of $Re_x$ that are much above 10 $^5$ . Therefore, the Blasius result represents an ideal goal, yet it practically unobtainable for most applications. If $\delta$ is defined as the value where $u=0.99U$ then for a laminar boundary layer

(13) $\begin{equation*} \delta \simeq 5.2 x Re^{-0.5} \end{equation*}$

and for a turbulent boundary layer

(14) $\begin{equation*} \delta \simeq 0.37 x Re^{-0.2} \end{equation*}$

Turbulent Boundary Layer

The principles surrounding the development of a turbulent boundary layer have already been discussed, a turbulent boundary layer being distinctive because it has more mixing between the fluid layers. However, the structure of a turbulent boundary layer is more complicated, and there are no theoretical results like the Blasius solution. A turbulent boundary layer has been well-studied experimentally, but it is not easy to develop a simple, functional relationship for the velocity profile. Of course, this approach makes the modeling and analysis of turbulent boundary layers much more complicated than for laminar boundary layers. However, various approximations for the turbulent boundary layer profile can be used.

Among the simplest and the best-known approximations is the power-law profile, which is expressed as

(15) $\begin{equation*} \frac{u}{U_{\infty}}} = \left(\frac{y}{\delta}\right)^{1/n} \end{equation*}$

where the value of $n$ is a constant that depends on the Reynolds number. It is found that the value of $n$ tends to increase with increasing Reynolds number to give agreement with measured profiles of turbulent boundary layers. However, for many flows, the use of $n = 7$ is sufficient to represent a turbulent boundary layer profile called the “one-seventh power-law.” However, a disadvantage of the power-law profile is that it cannot be used to calculate the shear stress exactly at the wall (i.e., at $y = 0$ ) because it gives an infinite velocity gradient. Notice also that a turbulent boundary layer will be thicker than a laminar one, a point previously made, because of the larger extent of mixing in the flow.

Example #2 – Skin Friction from Laminar & Turbulent Boundary Layers

Consider two boundary layer profiles at a wall, one laminar and one turbulent. The laminar profile is given by $u/U=2(y/\delta)-(y/\delta)^2$ and the turbulent profile by $u/U=(y/\delta)^{1/7}$ where $U$ is the edge velocity (external velocity) and $\delta$ is the boundary layer thickness. Calculate the shear stress on the wall in each case where $\delta$ for the turbulent boundary layer is twice that of the laminar boundary layer and explain why they differ.

The shear stress, $\tau_w$ , on the wall in each case is given by

$\tau_w = \mu \left( \frac{du}{dy} \right) = \mu \left(\frac{U}{\delta} \right) \left( \frac{d (u/U)}{d (y/\delta)} \right) \mbox{ as $(y/\delta)\rightarrow 0$}$

So, for the laminar boundary layer.

$\frac{d (u/U)}{d (y/\delta)} = 2 \left(1 - (y/\delta) \right) = 2 \mbox{ as $(y/\delta)\rightarrow 0$}$

and for the turbulent boundary layer

$\frac{d (u/U)}{d (y/\delta)} = (1/7) (y/\delta)^{-6/7} = 53.25 \mbox{ as $(y/\delta)\rightarrow 0.001$}$

where in the latter case, the gradient has been evaluated at a short distance away from the wall because the gradient would otherwise become infinite as $(y/\delta)\rightarrow$ 0. If $\delta$ for the turbulent boundary layer is twice that of the laminar boundary layer, then the latter value will be reduced by about 1.81 to 29.39.

It is clear then that the turbulent boundary layer produces a much higher skin friction coefficient on the wall, for given values of $U$ and $\mu$ . Hence, it will also produce a much higher drag on a body over which such a boundary layer may exist.

Visualizing the Boundary Layer

Boundary layers are typically very thin regions next to a surface, so they are fairly challenging to visualize using measurement techniques. Nevertheless, standard flow visualization methods used in the wind tunnel environment can reveal the signature and perhaps specific boundary layer properties. Flow visualization may be divided into surface flow visualization and off-the-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.

An example illustrating the effects of the two boundary layer states on an airfoil at a low angle of attack and relatively low Reynolds number is shown in the figure below. In this case, the flow was visualized in a wind tunnel by employing a surface oil flow technique in which light mineral oil is mixed with a white paint powder, and the mixture is applied to the surface with a brush. The shear stress on the oil mixture from the boundary layer creates a pattern, with regions of lower skin friction leaving behind more significant oil accumulations.

This image is courtesy of Michael Selig at the University of Illinois, obtained with permission. — Surface flow visualization on a airfoil at low angle of attack showing the transition from a laminar to turbulent boundary layer. Eppler 387 airfoil, $Re=350,000, \alpha=2^{\circ}$ .

Surface flow visualization on a airfoil at low angle of attack showing the transition from a laminar to turbulent boundary layer. Eppler 387 airfoil, $Re=350,000, \alpha=2^{\circ}$ .

For these conditions, the forward 60% of the airfoil can be interpreted to have a laminar boundary layer; the low surface shear stress here allows the oil to accumulate, especially near the mid-chord where the boundary layer approaches the transition point. Downstream of the transition, the boundary layer state is fully turbulent, the higher shear stresses causing more of the oil on the surface to be removed. Notice that the transition region is a short region where the flow is separated, which is indicative of a laminar separation bubble. However, it must be remembered that what is observed on the surface is not always indicative of what is happening in the external flow.

Skin Friction Drag on an Airfoil

Estimating the skin friction drag on bodies with a turbulent boundary layer is an important problem in aerospace engineering. While the problem is generally complex for a complete airplane or other flight vehicles because of the complex surfaces and the highly three-dimensional boundary layers that subsequently develop, the process can be illustrated for a two-dimensional airfoil. An airfoil is reasonably thin and flat, so at least at low angles of attack, the airfoil can be approximated by a flat plate.

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

(16) $\begin{equation*} C_d = \frac{2}{c} \int_0^c c_f \, dx = 1.328 \, Re^{-0.5} \end{equation*}$

where the factor “2” is needed because the plate has upper and lower surfaces, his latter result being plotted for reference in the figure below.

Viscous drag of a flat plate compared to minimum drag coefficients of several airfoils.

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

(17) $\begin{equation*} c_{f} = 0.0583 Re_{x}^{-0.2} \end{equation*}$

This latter result is obtained by assuming the velocity profile corresponds to a 1/7th power law profile, as given previously. If the plate was 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

(18) $\begin{equation*} C_d = 0.1166 \, Re_c^{-0.2} \end{equation*}$

The validity of the latter expression is generally limited to a $Re$ range between $10^{5}$ and $10^{9}$ . Below $Re = 5 \times 10^{5}$ the boundary layer can normally be assumed to be laminar, that is, unless the pressure gradients are particularly adverse, which will slow the flow near the wall and cause the boundary layer to separate.

These 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., above $10^6$ . In this case, it will be sufficient to assume that

(19) $\begin{equation*} C_d = C_{d_{\rm ref}} \left( \frac{Re}{Re_{\rm ref}} \right)^{-0.2} \end{equation*}$

where $Re_{\rm ref}$ is the reference chord Reynolds number for which a reference value of drag $C_{d_{\rm ref}}$ 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, which was suggested by McCroskey,

(20) $\begin{equation*} C_d = 0.0044 + 0.018 Re^{-0.15} \end{equation*}$

offers an improved correlation at the higher chord Reynolds numbers, which will be typical of those found on larger aircraft and at higher flight speeds. At very low Reynolds numbers, say below $10^5$ , the laminar boundary layer separates more readily. The drag coefficients of typical airfoils larger than either laminar or turbulent boundary layer theory would suggest.

Airfoil behavior in this Reynolds number regime is important for many classes of small-scale air vehicles, i.e., MAVs. In this case, it can be seen in the figure that changing the scaling coefficient from 0.2 to 0.4 in Eq. ref{cdcorel} gives a good approximation (fit) based on the results shown over a lower range of chord $Re$ . However, the drag of airfoils in this regime tends to be less predictable, especially at other than shallow angles of attack, because the boundary layers are thicker, often with laminar separation bubbles, and the onset of flow separation occurs more readily.

Effect of Surface Roughness

The development of a boundary layer is profoundly affected by pressure gradients but also by surface roughness. For example, the transition from a laminar to a turbulent boundary layer flow is often caused prematurely by the effects of surface roughness, as shown in the figure below, which acts to enhance the mixing of the flow in the lower layers of the boundary layer and so to produce turbulence more quickly. However, it should be appreciated that a laminar boundary layer is so thin that it takes only a tiny amount of roughness to initiate the transition process.

Surface roughness will cause mixing in the lower layers of the boundary layer and cause a laminar boundary layer to become turbulent prematurely.

As this turbulence develops near the surface, it mixes up through the higher layers in the boundary layer. As a result, it quickly progresses through the entire boundary layer from top to bottom, except immediately near the wall. Surface roughness can be present on a surface for various reasons, including just the “in-service” use of an aircraft, such as average abrasion at the leading edge of a wing, which will increase the drag of that wing. However, in some cases, such as with golf balls, surface roughness can have beneficial effects in that the transition to a turbulent boundary layer delays flow separation and reduces the pressure drag on the ball.

Boundary Layer Flow Control

There is much ongoing research into “laminar” flow wings and active flow control technology to improve aerodynamic efficiency. Control over the natural turbulent boundary layer that forms on an aircraft by keeping it laminar could reduce skin friction drag. Such an approach could reduce aircraft drag by up to an order of magnitude and significantly reduce fuel consumption, especially for high-speed aircraft.

NASA has tested a modified F-16 with a porous glove over the wing, as shown in the photograph below. The glove is the black region on the left wing. The glove contained more than 10 million tiny holes drilled by lasers and connected to a suction system. This approach removed a portion of the slower part of the boundary layer close to the surface, extending the laminar flow region across the wing. However, there are many challenges in developing practical boundary layer control systems for aircraft, including weight, reliability, maintainability, and costs.

The Supersonic Laminar Flow Control (SLFC) experiment on a modified F-16 aimed to reduce drag by using boundary layer suction control.

Other Properties of Boundary Layers

Besides the velocity profile $u = u(y)$ or $u/V_e = f(y/\delta)$ in the boundary layer, there are three other parameters that engineers use to quantify the characteristics of boundary layers, namely: 1. The displacement thickness; 2. The momentum thickness; 3. The shape factor.

1. The displacement thickness of a boundary layer is given the symbol $\delta^*$ and is defined by

(21) $\begin{equation*} \delta^* = \int_0^\delta \left( 1 - \frac{u}{V_e} \right) dy \end{equation*}$

the idea of which is illustrated in the figure below. Therefore, the displacement thickness of a boundary layer is equivalent to the height or thickness corresponding to the reduction in mass flow associated with the boundary layer profile, i.e., what is, in effect, a mass flow deficit.

The concept of displacement thickness of a boundary layer, which is equivalent to a reduction in mass flow rate by the presence of the boundary layer.

2. The corresponding momentum thickness of the boundary layer, which is given the symbol $\theta$ , is defined by

(22) $\begin{equation*} \theta = \int_0^\delta \frac{u}{V_e} \left( 1 - \frac{u}{V_e} \right) dy \end{equation*}$

which again has physical meaning because it gives a measure of the reduction in momentum in the flow from the development of the boundary layer, i.e., a quantity that gives a measure of the momentum deficit in the flow.

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

(23) $\begin{equation*} H = \frac{\delta^*}{\theta} \end{equation*}$

The shape factor is often used as an indicator to help characterize the state of boundary layer flows and evaluate their propensity to stay attached or to separate from surfaces. Notice that because

(24) $\begin{equation*} 1 - \frac{u}{V_e} > \frac{u}{V_e} \left( 1 - \frac{u}{V_e} \right) \end{equation*}$

then the value of $H$ is always greater than unity.

It has been found from experiments that the values of $H$ take on ranges of values depending on the flow state. For example, for laminar flows then $H \approx 2.5$ and for fully turbulent flows then $H \approx 1.5$ . As a boundary layer approaches the point of separation from a surface, then the values of $H$ typically increase to 4 or more. Therefore, to calculate the development of a boundary layer over a wing, then as the value of $H$ increases to around 4, it would be anticipated that the boundary layer is about to separate from the surface. Therefore, the shape factor is one parameter that can be used to help design a contour of an aerodynamic surface so that the flow remains more attached than it might otherwise.

Summary & Closure

Understanding the characteristics of boundary layers as they develop over solid surfaces is essential for understanding the aerodynamic characteristics of all types of bodies, both lifting and non-lifting. Besides the effects of skin friction, which is a significant drag source on most flight vehicles, boundary layers can separate from surfaces and cause deleterious effects on the resulting aerodynamics. While laminar boundary layers are the easiest to understand, the reality of most flows about flight vehicles is that their boundary layers are fully turbulent. While studying turbulent boundary layers is a technical field within itself, the essential characteristics of turbulent boundary layers and how they behave are, for example, critical to understanding the aerodynamic characteristics of airfoils, wings, and other shapes.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Consider two fully developed turbulent boundary layers that are otherwise identical, but one has twice the thickness of the other. What will be the relationship between the wall stresses in each case?
Think about how to quantitatively measure the wall stress exerted by a boundary layer on a surface, such as on a body being tested in a wind tunnel.
How might surface roughness affect the development of a turbulent boundary layer?
Explain why laminar boundary layers are more likely to separate than turbulent boundary layers when experiencing the same adverse pressure gradient.
Write a MATLAB program to calculate the solution to the Blasius profile and plot the thickness, displacement thickness, and momentum thickness for $Re_x$ = 1,000.

Other Useful Online Resources

To learn more about boundary layers, check out some of these additional resources:

Some good information on boundary layers from Wikipedia.
Great introductory video on the concept of a boundary layer.
What is a Boundary Layer? Laminar and Turbulent boundary layers explained.
Fluid Mechanics, The Boundary Layer; SAFL Film No. 56
NASA’s technical information page on boundary layers.
Basic fluid dynamics video lecture on boundary layers from YouTube.
An excellent old film on boundary layers sponsored by NSF.
YouTube video: Turbulent Flow is MORE Awesome Than Laminar Flow!

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Introduction to Aerospace Flight Vehicles by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/ https://doi.org/10.15394/eaglepub.2022.1066.n22