Introduction

Turbulent or mixing flows can occur in all types of fluids. Turbulence arises when disturbances in a flow are amplified by nonlinear inertial effects faster than they are damped by viscosity. The Reynolds number, a parameter named after the pioneering work of Osborne Reynolds, is the primary nondimensional parameter governing the tendency of a flow to transition from laminar to turbulent motion. For internal flows such as flow through a smooth circular pipe, the flow usually remains laminar for , becomes transitional over roughly , and is usually turbulent for . For flows over smooth surfaces such as airfoils and wings, transition often occurs at a local Reynolds number on the order of to , based on the distance from the leading edge. However, transition does not occur at a single universal critical Reynolds number because it also depends on disturbance levels, surface roughness, pressure gradients, geometry, acoustic noise, wall temperature, and other environmental influences.

Velocity gradients and shear flows provide the conditions under which turbulent fluctuations can extract energy from the mean flow, while viscosity ultimately dissipates turbulent kinetic energy at the smallest scales. In particular, strong shear flows are produced near solid boundaries because of the no-slip condition at the wall, as shown in Figure 1. Shear layers also occur at interfaces between fluid streams with different velocities, for example, at the edges of wakes and jets.

A random, irregular motion of the fluid elements is a characteristic of all turbulent flows. In fluid dynamics, the term “turbulent flows” is often used in place of “turbulence,” though the terms are often used interchangeably in most contexts. The creation of turbulent flows, or turbulence, is often referred to as a “chaotic” or stochastic process. It involves changes in local flow velocity and pressure fluctuations from the formation of vortices and eddies, as shown in Figure 2. In a laminar flow, the fluid particles follow the streamlines precisely, as indicated by the linear dye trace in the laminar region. Turbulent flow involves the superposition of the effects of eddies of various sizes onto the mean flow. The dye’s path, therefore, is influenced by both the bulk or mean flow (i.e., the time-averaged or mean streamlines) and the location and strength of the eddies. Larger eddies can transport the dye across the mean streamlines, thereby introducing considerable mixing into the flow. Additionally, smaller eddies will further mix the dye, causing it to diffuse quickly.

One characteristic feature of turbulent flows is their multiscale nature: larger eddies transfer energy to smaller eddies, which in turn transfer energy to still smaller ones through the turbulent energy cascade. This cascade was described qualitatively by Lewis Fry Richardson and later placed on a statistical foundation by Kolmogorov, who defined the Kolmogorov microscales at which viscosity dissipates turbulent kinetic energy. Therefore, turbulent flows around airplanes consist of a hierarchy of eddy scales, ranging from the order of the wing chord down to much smaller dissipative scales determined by the Reynolds number, viscosity, and local rate of turbulent-energy dissipation. The smaller-scale eddies, such as those that persist after a disturbance has ended, eventually spin down and dissipate their rotational and translational energy as heat through viscosity-induced shear forces. Richard Feynman described “turbulence” as one of the most important unsolved problems in classical physics.^[1]. However, explaining the nature of turbulence is probably the most important unsolved problem in mathematics, not physics, because of the need to represent it in some form of equation.

Engineers continue to seek mathematical models of turbulent flows, such as those within the framework of the Navier-Stokes equations, because the behavior of these flows is integral to solving many real-world problems. Turbulent flow modeling is crucial for improving aircraft performance, forecasting weather conditions, and gaining a deeper understanding of chemical reactions and energy production. Turbulence plays a significant role in various aerodynamic problems, as well as in combustion, heat transfer, fluid-structure interactions, and noise generation. Turbulent flows increase drag on aircraft, primarily because of the higher skin friction associated with turbulent boundary layers. Turbulent flows also produce thicker boundary layers and higher skin-friction drag. Their effect on pressure drag depends on the flow situation; in some cases, turbulent mixing can delay separation and reduce pressure drag, while in others it can increase overall drag. However, turbulent flows are also harnessed in engineering applications, such as fuel mixing in engines, combustion, and efficient heat transfer in industrial processes. Turbulent flows can also be used to prevent flow separation from lifting surfaces where it would otherwise occur. This is sometimes achieved using vortex generators, a form of flow control.

Visualizing Turbulent Flows

In the 16th century, the great polymath Leonardo da Vinci created sketches of turbulent flows, two of which are shown in Figure 3 below, taken from his famous notebooks, known as codices. He described the existence of whirlpools and “whorls” or eddies and wrote, “Observe the motion of the surface of the water, resembling that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the whorls; thus the water has whorling motions, one part of which is from the principal current, the other to random and reverse motion.”^[2] In other words, Leonardo da Vinci explains the essence of a turbulent flow, with whorls and vortices of various sizes formed, causing some level of ordered three-dimensional fluid motion, usually referred to as the bulk flow, and some level of disordered or random motions, which is now referred to as “turbulence.”

Da Vinci also noted, “The small whorls are almost numberless, and large things are rotated only by large whorls and not by small ones, but both turn small whorls.” Da Vinci also states, “So moving water strives to maintain the course under the power which occasions it and, if it finds an obstacle in its path, completes the span of the course it has commenced by a circular and revolving movement.” These profound observations, perhaps the first documented visualizations of turbulent flow, have since inspired those pursuing mathematical descriptions of turbulent flow.

The visualization image in Figure 4 illustrates another example of the fascinating natural phenomenon of turbulent flow. The refractive or schlieren effects originate from a heat source (a candle, in this case), which initially causes the heated air to rise vertically as a smooth, laminar plume. At some point downstream, depending on the Reynolds number, the flow transitions to turbulence. This process is characterized by the formation of relatively large eddies, which increase the mixing levels of the flow. Although the process is inherently random, it nevertheless exhibits underlying order and structure, with patterns that, in their complexity, can be seen as possessing a certain beauty.

Notice from the image above that mixing proceeds very quickly as the flow develops, becoming increasingly turbulent, and that the thermal jet’s boundaries rapidly expand. The expansion occurs because turbulent mixing entrains surrounding fluid into the rising thermal jet, increasing its mass flow rate while slowing the average upward velocity and causing the jet to spread as it develops downstream. Images such as these confirm that turbulent flows involve energy transfer across various physical scales: the initial larger-scale eddies, on average, break down into smaller-scale eddies. This energy cascade continues until it reaches the smallest scales, where the energy is eventually dissipated by viscosity. Recall that viscosity is the property of a fluid that resists deformation, so it acts as a dissipative energy mechanism. Therefore, the kinetic energy in the turbulence is ultimately converted into heat through shear deformations and viscous dissipation.

Reynolds Decomposition

As previously mentioned, turbulence is best described as a stochastic process, although it also exhibits deterministic chaotic behavior. While the underlying Navier-Stokes equations governing fluid motion are deterministic, the complexity and range of interacting scales in turbulence make the exact prediction of every detail impractical. Instead, turbulence is treated statistically, relying on averages, fluctuations, and probability distributions to describe its behavior. Unlike chaos, which is deterministic but unpredictable because of a sensitivity to initial conditions, turbulence involves random-like fluctuations that exhibit statistical regularities over time. These stochastic characteristics enable the formulation of meaningful statistical descriptions of turbulent flows.

A technique used by engineers to model turbulent flows is the Reynolds decomposition. In this method, an instantaneous flow property such as a velocity, , is decomposed into a mean value of velocity, , with an associated fluctuating component, , i.e., the flow velocity can be expressed as , as shown in Figure 5. Each fluctuation of is associated with the instantaneous motion of one or more eddies in the flow, which may cause the local velocity to increase or decrease depending on the velocity induced at the measurement point. Reynolds decomposition is a critical concept in mathematical models of turbulent flows, yielding a set of governing equations known as the Reynolds-Averaged Navier-Stokes (RANS) equations.

Reynolds decomposition can also be extended to three dimensions, in which each velocity component or scalar quantity can be decomposed separately into its mean and fluctuating components. This approach yields a more comprehensive description of three-dimensional turbulent flow, facilitating a deeper understanding of the fluid transport phenomena within it. For example, if Reynolds decomposition is applied to the three components of the flow velocity, then

(1)  

where the , , and terms are the mean values of the flow velocity, and the , , and terms are the associated fluctuating components from the mean, respectively. Notice that , , and , as well as , , and , are functions of time and space, but the averages , , and are functions of space only for a statistically steady flow.

The mean values can be obtained by averaging the flow properties over a long period, say , so that for a given spatial location in the flow, say , then

(2)  

The duration of required to obtain the desired information depends on the flow. The general rule is that must be long enough for statistical convergence, which often means as long as practically possible within the times and constraints of any experiment. Formal convergence studies may also be necessary to ensure that the results fall within accepted statistical bounds. Therefore, the turbulent parts of the flow velocities are

(3)  

from which the root mean square (RMS) or “strength” of the turbulent flow can be defined as

(4)  

In practice, these averaged values are derived from discrete measurements (i.e., flow measurements taken at discrete time intervals), so summations replace integrals. For a discretely sampled time history of values on statistically correlated three-component measurements of velocity , then performing Reynolds decomposition gives the mean velocity components as

(5)  

where is the number of contiguous measurements to ensure statistical validity. The value of must be sufficiently large to obtain statistical convergence, typically requiring thousands of samples. Some turbulent events may occur sporadically and be brief, so contiguous measurements are essential to fully capture their behavior. For unsteady flows, especially periodic ones, phase averaging may be necessary. In this process, the mean flow values are obtained by summing the measured values across multiple in-phase cycles, triggered by a periodic or other defined event.

Reynolds’ Rules of Averaging

The rules of statistical averaging are crucial for formulating the RANS equations and enabling their solution. If and are fluctuating quantities varying in time such that

(6)  

Then, specific rules known as the Reynolds averaging rules listed below will apply. These rules follow statistical precedents from mathematics, and it is a straightforward exercise to prove that they follow from Eq. 6.

• The average of the sum is the sum of the averages, i.e.,

(7)  

• The average of the turbulent values is zero, so

(8)  

• Constants do not affect and are not affected by averaging, i.e.,

(9)  

• The average of the product is the product of the averages plus the average of the product of the fluctuations

(10)  

• Averaging products also leads to

(11)  

• In terms of the derivatives found in the Navier-Stokes equations, the average of the time or space derivative of a quantity is equal to the corresponding derivative of the average, i.e.,

(12)  

Navier-Stokes Equations

Recall that the incompressible form of the Navier-Stokes equations can be written as

(13)  

where is the vector Laplacian. Notice that

(14)  

For an incompressible Newtonian flow, , so the divergence of the viscous stress tensor reduces to when the viscosity is constant. The viscous stresses include both normal and shear components, all of which depend on gradients of the velocity components. The Navier-Stokes equations can also be written in the form

(15)  

where is the kinematic viscosity; this latter form is the most common way of writing the Navier-Stokes equations.

Notice that there is nothing in the assumptions and derivation of the Navier-Stokes equations to suggest that they are limited to laminar flows. In principle, they also govern turbulent flows, but in practice, the instantaneous initial and boundary conditions needed to resolve all turbulent scales cannot usually be specified. This difficulty can be addressed by using Reynolds averaging. Proceeding by expanding out the substantial derivative in Eq. 13 using

(16)  

gives

(17)  

and in terms of the scalar components, this becomes

(18)  

For steady flows, the equations are reduced to

(19)  

and in terms of the scalar components, they are

(20)  

Including Turbulent Flow Effects

Reynolds’ decomposition and the averaging rules can now be applied to the continuity and momentum equations. The result will be the Reynolds-averaged Navier-Stokes (RANS) equations. This form of the Navier-Stokes equations serves as the basis for most computational fluid dynamics (CFD) simulations of complex flows.

Continuity

For a steady, incompressible flow, the continuity equation is

(21)  

So, in terms of the Reynolds decomposition, then

(22)  

Taking only the mean value of Eq. 22 gives

(23)  

and subtracting Eq. 23 from Eq. 22 gives

(24)  

Therefore, the mean or bulk flow in terms of Eq. 23 and the fluctuating flow in terms of Eq. 24 both satisfy continuity.

Momentum

The momentum terms can now be considered. A turbulent flow is instantaneously unsteady even when its mean properties are statistically steady. Therefore, the Reynolds averaging must begin with the unsteady momentum equation in Eq. 18. After averaging, the mean time-derivative term vanishes for a statistically steady flow. Multiplying Eq. 21 by and adding it to the convective terms in the component of Eq. 18 gives

(25)  

Therefore, the mathematics leads to

(26)  

The pressure is , and the three velocities are , , and , respectively. Therefore, assuming a steady flow and taking only the mean values gives

(27)  

Using Reynolds averaging gives

(28)  

Then subtracting times Eq. 23 from the first term in parentheses in Eq. 28 gives

(29)  

or

(30)  

The corresponding equations for the and directions are

(31)  

and

(32)  

which can finally be written in the vector form

(33)  

which are known as the Reynolds-averaged Navier-Stokes (RANS) equations. Recalling the original form of the Navier-Stokes equations, i.e., their “laminar” form without the turbulent flow terms, then

(34)  

so it can be seen that Eq. 33 differs from the original form of the Navier-Stokes equation in Eq. 34 because of the additional Reynolds-stress divergence term, , on the right-hand side. The tensor is referred to here as the Reynolds-stress tensor. The stress tensor, which is a matrix of stress terms and is illustrated schematically on the elemental control volume in Figure 6, accounts for the additional effects of turbulent flow on the momentum transfer and can be expressed as

(35)  

The terms , , and , etc., contribute additional stresses in the flow over and above those caused by viscosity alone. Notice that the matrix’s diagonal terms represent the normal stresses, and the off-diagonal terms are the shear stresses.

For the Reynolds-stress tensor, the off-diagonal terms are symmetric because the products of the fluctuating velocity components commute before averaging, i.e.,

(36)  

Therefore, of the nine Reynolds-stress components, only six are distinct. This symmetry should not be confused with the separate physical argument for the symmetry of the molecular stress tensor, which follows from angular-momentum balance for a fluid element.

The preceding Reynolds-averaged equations are exact consequences of averaging the incompressible Navier-Stokes equations, but they are not closed because the Reynolds stresses introduce additional unknowns. However, the appearance of the six Reynolds stresses means that the number of unknowns has, in effect, increased to ten (pressure, three components of the flow velocity, and six Reynolds stress terms), while only four equations are available, so the properties cannot be uniquely solved. Further progress requires additional equations to relate the Reynolds stresses to the flow velocity components. Indeed, the fundamental problem in turbulence modeling within the RANS framework is to find six additional mathematical relations to close the system of four equations.

Interpretation of Reynolds Stresses

The additional terms that appear in the Reynolds-averaged Navier-Stokes equations, i.e., the Reynolds stresses, represent the transport of momentum by turbulent velocity fluctuations. These stresses arise because fluid parcels moving across a mean velocity gradient carry momentum that differs from that of the surrounding fluid, thereby producing an additional apparent stress.

An order-of-magnitude estimate illustrates their importance. In a boundary layer, the turbulent shear stress is

(37)  

whereas the viscous shear stress is

(38)  

Away from the wall, the velocity fluctuations scale with a characteristic velocity of the flow, so that , giving

(39)  

In contrast, the viscous stress scales as

(40)  

where is a characteristic velocity and is the boundary-layer thickness. For high Reynolds number flows, then

(41)  

so that turbulent stresses dominate over viscous stresses throughout most of the boundary layer, except in a thin region adjacent to the wall.

This result highlights the dominant role of turbulent momentum transport and shows that the Reynolds stresses must be modeled to obtain a closed set of equations. The development of such models forms the basis of practical turbulence modeling approaches.

Eddy Viscosity Models

Notice that the viscous and turbulent shear terms can be written in the form

(42)  

where

(43)  

The term involving is negligible in comparison to under boundary-layer scaling. Therefore, the sum of a laminar shear stress component and a turbulent shear stress component, in effect, replaces the “laminar” viscous stresses. This means that

(44)  

where is referred to as the eddy viscosity, i.e., the extra effective viscosity caused by turbulence. Therefore, turbulence models that represent variations in eddy viscosity are referred to as eddy-viscosity models. In an eddy viscosity model, higher levels of turbulent stresses are produced by augmenting the otherwise effects of pure molecular viscosity with a turbulent or “eddy” viscosity coefficient.

This latter assumption is known as the Boussinesq Hypothesis, after Joseph Boussinesq, although he never explicitly referred to it as such. However, the assumption, or “hypothesis,” is reasonable because it posits that the exchange of turbulent momentum in the eddy-cascading process is analogous to the effects of molecular viscosity, as illustrated in Figure 7. Experiments and other observations of turbulent flows support such behavior. Therefore, the Boussinesq hypothesis implies that turbulent flow can be modeled as an effective total viscosity that varies with local flow conditions.

The Boussinesq hypothesis is a widely used assumption in fluid mechanics that relates the eddy viscosity, a measure of turbulent momentum transport, to the mean rate-of-strain tensor in the Navier-Stokes equations. In its more complete tensor form, the viscous and modeled turbulent stress terms are proportional to the symmetric part of the mean velocity-gradient tensor. For many introductory engineering applications, especially for constant-property incompressible flows, this behavior is often represented using the simplified effective-viscosity form

(45)  

where is interpreted as an effective total viscosity acting on the mean velocity field. This form emphasizes that the effective viscosity is generally a locally modeled quantity, not a constant that can always be taken outside the differential operator.

Experience has shown that the preceding approach to modeling turbulence within the framework of the Navier-Stokes equations is reasonably practical for certain classes of flow problems, particularly those involving homogeneous mixing, dimensional symmetry, and relatively isotropic turbulence.

Turbulence Scales & Turbulent Kinetic Energy

Eddy viscosity models belong to a class of so-called “turbulence models” or “turbulence closure models,” which provide a mathematical basis for solving both the mean flow field and the turbulent flow quantities. Many different turbulent flow models are available, particularly for use in computational fluid dynamics (CFD). Choosing the most suitable turbulence model requires careful consideration of the specific problem. The best way to represent turbulent viscosity and select the most suitable turbulence model depends on several factors, including the specific flow application, the desired level of accuracy, and the available computational resources.

Deciding which turbulence models are likely to provide good predictive capability depends on understanding the length and time scales of the specific flow problem at hand. Turbulent flows exhibit a wide range of spatial scales, each of which plays a vital role in the flow’s overall behavior. The spatial scales of turbulence can be broadly classified into three categories: large, intermediate, and small eddies, as illustrated in Figure 8. Therefore, there is no “one size fits all” turbulence model. The different scales of turbulence interact in complex ways, and a wide range of phenomena, including vorticity, turbulence intensity, and Reynolds stress, characterize their behavior. Modeling these scales is crucial for predicting turbulent flow behavior, a critical challenge in turbulence modeling.

1. Large-scale eddies are the most significant structures in a turbulent flow. They transport bulk momentum and energy and can cause substantial fluctuations in the local flow properties.
2. Intermediate-scale eddies: These eddies are smaller than the large-scale eddies but larger than the smallest scales. They are essential in transferring energy from the large-scale eddies to the smallest scales through the “energy cascade” process.
3. Small-scale eddies are the smallest structures in the turbulent flow. Their dimensions depend on the fluid properties, Reynolds number, and characteristic scales of the flow. At the smallest dissipative scales, viscosity converts turbulent kinetic energy into thermal energy through viscous shear.

Eddy Turnover Time

In turbulent flows, multiple velocity scales are relevant to understanding flow behavior. These velocity scales are related to the flow’s characteristic length and time scales, reflecting the various physical phenomena that may occur. They can also be used to classify the overall behavior of the flow. One important time scale in turbulent flows is the eddy turnover time. This timescale is related to the size of the eddies in the flow and reflects how quickly they break down and reform. The eddy turnover time, , is given by

(46)  

where is the characteristic length scale of the eddy, and is the characteristic velocity scale of the eddy. It represents the characteristic time over which an energy-containing eddy evolves or turns over. This timescale is important in the context of the larger turbulent cascade, where energy is transferred from larger to smaller eddy scales.

Turbulence Intensity

Turbulent intensity measures the level of turbulence in a fluid flow, typically expressed as a percentage of the mean flow property. It is defined as the root-mean-square (rms) of the fluctuating property divided by the mean value. For the streamwise component of the velocity fluctuation, the streamwise turbulence intensity is

(47)  

where is the rms of the fluctuating flow component, and is the mean flow velocity. The mean flow velocity is often taken as the freestream reference, i.e., or . A three-component turbulence intensity may also be defined as

(48)  

where is the chosen reference mean velocity.

Turbulence intensity provides one measure of the “strength” of turbulence in a fluid flow. Higher intensity values generally indicate more turbulent flow with larger velocity fluctuations. Measuring turbulence intensity employs techniques such as hot-wire anemometry (HWA) and particle image velocimetry (PIV). Hot-wire anemometry can provide high-temporal-resolution point measurements, whereas PIV provides spatially resolved velocity measurements, with temporal resolution depending on the imaging and illumination system. Medium turbulence intensities (about 5%) are found in propellers, fans, and fully developed internal flows. High-speed flows, such as those inside turbines and jets, exhibit greater than 10% turbulence relative to the mean flow.

Turbulent Kinetic Energy

The turbulent kinetic energy, , also known as TKE, is a crucial parameter in fluid dynamics, often used to characterize turbulence intensity in a flow. The equation that defines it is

(49)  

TKE is generated when a fluid encounters surfaces, obstacles, or other changes in velocity, with turbulent eddies converting mean-flow energy into turbulent energy.

The numerical values of TKE are often used as a validation metric for comparing CFD simulations with flow measurements in wind tunnels or with other CFD models. While not the only validation metric, the TKE provides an excellent basis for assessing the performance of turbulence models. However, uncertainty quantification for measurements and calculations is needed to ensure valid comparisons and meaningful decisions.

Turbulence Spectra

The turbulent kinetic energy is distributed over a wide range of length scales. This distribution is described by the energy spectrum, , which is usually plotted against the wave number on logarithmic axes, as shown in Figure 9. The wave number is given by

(50)  

Notice, therefore, that bigger eddies have lower wave numbers, and the smaller eddies have larger wave numbers. The basic idea of this form of presentation is to show the contribution of the turbulent kinetic energy at a particular wave number, , which results from eddies with characteristic diameter, , and corresponding wavelength . The use of the wave number stems from the mathematical development of Kolmogorov’s universal equilibrium theory, which characterizes the distribution of turbulence across different length scales and times.

The lower left limit of the energy plot corresponds to the length scales at which turbulence is initially generated, i.e., from the biggest eddies. As new eddies form and gain energy from the bulk flow, they generate smaller eddies, which in turn generate even smaller ones in a process known as cascading. The spectrum of spatial or temporal eddy sizes then reaches a “fully developed” state, in which energy is continuously fed from the mean flow into increasingly smaller eddies. This so-called inertial region is the range of scales over which turbulent kinetic energy is transferred from larger eddies to smaller eddies with little direct production or viscous dissipation. In high-Reynolds-number turbulence, the scaling in this region is approximately independent of viscosity, although the largest energy-containing eddies and the smallest dissipative eddies still depend on the flow conditions and Reynolds number.

The slope in this inertial region is , as described by the Kolmogorov theory, as shown by the dashed line in Figure 9. This latter behavior, characteristic of turbulent flows and occurring in many practical contexts, can also help prescribe the power spectra for turbulence modeling in computational fluid dynamics (CFD) simulations. The energy-cascading process eventually ends because the smaller eddies continue to break down until they become small enough for viscosity to dissipate their kinetic energy; the remaining heat is then dissipated.

Wall-Bounded Turbulent Flows

Turbulent boundary layers facilitate greater mixing between the fluid layers and bring higher-momentum fluid closer to the wall. Therefore, steeper velocity gradients exist near the wall, resulting in higher shear stresses in the fluid. This is why surfaces like airfoils and wings experience higher drag in turbulent boundary-layer flows than in laminar boundary-layer flows. Unfortunately, laminar boundary layers are easily disturbed and quickly develop into turbulent boundary layers, especially at the higher Reynolds numbers typical of actual flight vehicles. In fact, for most airplanes, except perhaps for high-performance sailplanes, laminar boundary layers exist only over very small parts of the leading edges of their wings before they quickly transition into turbulent boundary layers.

Turbulent Boundary Layer Equations

Applying the process of Reynolds decomposition of a turbulent flow gives an alternative subset form of the boundary layer equations, i.e.,

(51)  

which, like the RANS equations, yields Reynolds-stress quantities in terms of the fluctuating velocity components. Notice that the instantaneous pressure is also decomposed and then averaged, so the boundary-layer equations are written in terms of the mean pressure, .

Using the same approach as before, with the laminar form of the equation, where the orders of magnitude of the relevant terms in the boundary layer region at the wall were obtained, the form of the turbulent boundary layer equations becomes

(52)  

Again, it can be seen from a comparison with the “laminar” form of the boundary layer equations that the time-averaged values , , and have replaced , , and , respectively. Nevertheless, Reynolds stresses must be accounted for in the boundary layer. Reynolds stresses exhibit specific behavior in boundary layers, and appropriate turbulence models must be developed to predict their effects on wings and bodies. The ability to predict, and to do so for the right reasons, is fundamental to the design process.

Mixing Length Theory

All components of the turbulent, or Reynolds, stresses must approach zero at a solid boundary because the fluctuating velocity components must satisfy the no-slip and no-penetration conditions at the wall. Therefore, the eddy viscosity also approaches zero at the wall. Very close to the wall, the dominant shear stress is the molecular viscous stress associated with the mean velocity gradient. However, this statement does not imply that the wall shear stress is the same as that in a fully laminar boundary layer, because it is not; the entire turbulent boundary layer sets the near-wall mean velocity gradient. The boundary conditions at the wall are therefore expressed in terms of the mean velocity components, in the same manner as for a laminar flow. The very thin near-wall region in which viscous stresses dominate is called the viscous sublayer, also commonly known as the “laminar” sublayer, although the latter term should not be interpreted as implying a separate laminar boundary layer.

In boundary-layer problems, the Prandtl mixing-length model can be used. The assumption is that turbulence intensity varies with distance from the wall, , i.e., it is lower near the wall and higher farther away. As shown in Figure 10, the definition of mixing length implies that fluid elements migrating from level to level carry with them, on average, a velocity differential given by

(53)  

If it is also assumed that and are of the same order of magnitude, then the turbulent shear stress is

(54)  

The simplest assumption for the mixing length, , is given by

(55)  

where is the von Kármán constant. The mixing length is the distance over which fluid elements retain their original momentum as they move from one level to another within the turbulent boundary layer. Therefore, the eddy viscosity is given by

(56)  

or in terms of the kinematic viscosity

(57)  

The values of these latter two parameters depend on the type and class of flow problem and are not known a priori; therefore, mixing-length and eddy-viscosity models are empirical or semi-empirical closures that require calibration. They can be useful for prediction within the classes of flows for which they have been calibrated, but they are not universal turbulence models. In Prandtl’s mixing-length hypothesis, is introduced as a scale for turbulent eddies, representing the distance a fluid particle travels before losing its original characteristics and adapting to the local flow conditions. Here, typically varies with position within the boundary layer, typically as a function of distance from the wall in wall-bounded flows. In practice, values of are either:

1. Determined by calibration and fitting to experimental data.
2. Based on empirical models, such as those using wall distance in boundary layers.
3. Adjusted within computational models to best represent observed turbulent behavior.

Following the order of magnitude reduction approach done for laminar flows, one form of the turbulent boundary layer equation is

(58)  

where the unknowns are and for a given pressure gradient , and is the effective total kinematic viscosity. However, modeling turbulence in boundary layers is generally far more difficult because of its nonhomogeneous and nonisotropic nature. Much of the past and current research on turbulent boundary-layer flows focuses on mathematical descriptions of turbulence.

Law of the Wall

The mixing-length model yields another model for turbulent boundary-layer flow, known as the law of the wall. This law has been shown to hold for wall-bounded turbulent flows that develop under mild pressure gradients without flow separation. It helps to characterize and model the effects of turbulent flows near walls, which is crucial in many aerospace applications, such as in the design of airfoils, wings, and entire flight vehicles.

The law of the wall states that the velocity of a turbulent fluid flowing near a solid boundary exhibits a logarithmic relationship with the distance from the boundary. This type of relationship arises from the complex interactions between the fluid and the boundary, leading to turbulent velocity fluctuations, a phenomenon that has been experimentally verified. Mathematically, the law of the wall, which is an empirical construct, can be expressed as

(59)  

as shown in Figure 11, in terms of and , which are dimensionless parameters referred to as wall units. This is a classic presentation in boundary-layer theory, though it may initially seem somewhat nonintuitive. Nevertheless, this is one of the most famous empirically determined relationships for turbulent flows over solid boundaries.

The value of is a dimensionless mean velocity, defined as , where is the mean velocity parallel to the wall and is the friction velocity. The friction velocity is defined from the wall shear stress by

(60)  

or, equivalently,

(61)  

Despite its name, the friction velocity is not the actual velocity of the fluid at the wall. The fluid velocity at the wall is zero because of the no-slip condition. Instead, is a velocity scale that expresses the magnitude of the wall shear stress in velocity units. A higher wall shear stress yields a higher friction velocity. This velocity scale is especially useful because it allows near-wall velocity profiles obtained at different flow speeds, densities, and viscosities to be expressed in a common nondimensional form.

The wall shear stress is usually determined before calculating . If the mean velocity profile is resolved sufficiently close to the wall to determine its gradient at the surface, then

(62)  

so the wall shear stress follows from the velocity gradient at the surface. It may also be determined from the local skin-friction coefficient using

(63)  

where is the reference velocity. The value of may come from an analytical solution, an empirical correlation, a force measurement, or a computational fluid dynamics solution. In some experiments, is inferred by fitting measured near-wall velocity data to a known velocity profile. Once is known, the friction velocity follows directly from its definition.

The value of is defined as

(64)  

where is the distance from the wall and is the kinematic viscosity of the fluid. The parameter is a Reynolds number based on the friction velocity and the distance from the wall. It indicates the relative importance of inertial and viscous effects at a given distance from the surface. Therefore, the variables and provide a convenient way to describe the structure of the near-wall turbulent velocity profile.

The viscous sublayer relation is generally valid for , although the precise upper limit depends somewhat on the flow conditions. The transition, or “buffer layer,” occurs for . The logarithmic profile is valid farther from the wall in the logarithmic or overlap region of the turbulent boundary layer. The parameter is known as the von Kármán constant, with an accepted empirical value of 0.41. However, this value may change as higher-fidelity measurements in turbulent boundary layers become available. Finally, the value of is a constant that depends on the specific flow configuration and external boundary conditions and must also be determined empirically. Although semi-empirical, the law of the wall provides a valuable mathematical framework for understanding and predicting the effects of turbulent flows near solid boundaries.

Notice that the values of and depend on a knowledge of the wall stress, . There are various ways to measure and calculate . If flow velocities near the wall are available, a direct approach is to calculate the velocity gradient. However, achieving this with CFD requires a very fine grid near the wall, which may be infeasible because of grid size or computational limitations. Likewise, in experiments, obtaining sufficient flow-velocity measurements near a surface to determine a gradient can be challenging. Alternatively, indirect methods, such as a Stanton tube, Preston tube, or hot-film sensor, can be used to determine shear stress through calibration.

Turbulence Models

The Boussinesq hypothesis is commonly used in developing turbulence models to solve the Reynolds-averaged Navier-Stokes (RANS) equations and determine the value of . These models can be utilized to simulate turbulent flows in practical engineering applications. This hypothesis offers a straightforward, computationally efficient approach to incorporating turbulence effects into the Navier-Stokes equations, thereby enabling the prediction of mean-flow behavior and turbulence statistics. However, it must be recognized that the Boussinesq hypothesis is an approximation that represents the turbulent stresses using an effective eddy viscosity. This assumption works reasonably well for many attached shear flows but can be less reliable for separated flows, strong streamline curvature, secondary flows, and flows with large rotation or buoyancy effects.

In more complex turbulent flows, the Boussinesq hypothesis may need to be modified or replaced by more advanced turbulence models, especially considering that turbulence is often anisotropic. Various turbulence models are available to determine , each with its own assumptions and levels of complexity. Some commonly used turbulence models in RANS simulations include:

1. The Spalart-Allmaras (SA) turbulence model is a one-equation model that directly solves for the eddy viscosity. It is known for its simplicity and efficiency and is widely used, especially as a benchmark.
2. k-epsilon, or –, models solve two transport equations: one for the turbulent kinetic energy and another for the turbulent dissipation rate . Variations of this model include the RNG – and Realizable –.
3. k-omega, or –, models also solve two equations but use different variables: turbulent kinetic energy and specific dissipation rate . Variations include the SST – and BSL – models.
4. Reynolds Stress Models (RSMs) resolve the Reynolds stresses directly, providing more detailed information about the turbulence field. They are designed for use in flows with anisotropic turbulence, flow separation, and recirculation. However, RSMs are computationally more expensive than other models.

The choice of turbulence model depends on the specific characteristics of the simulated flow, the available computational resources, and the desired level of accuracy. However, the limited generality of such models across all flow types limits their ability to predict turbulent flow properties, particularly the development of turbulent boundary layers. Developing better turbulence models remains a goal in modern fluid dynamics, although these models are considered postdictive (i.e., developed after the fact) rather than predictive. It is common for engineers and researchers to perform sensitivity analyses using different turbulence models to assess their impact on simulation results and to compare them with available benchmark solutions. These benchmarks may encompass both experiments and other theoretical approaches.

Numerical Solution of the RANS Equations

Solving RANS equations numerically involves discretizing the governing equations into a set of algebraic equations that can be solved on a computer. The RANS equations consist of the continuity and momentum equations, which are typically solved using finite-volume or finite-difference methods.

1. As shown in Figure 12, the first step is to divide the computational domain into a fine grid (or mesh) of finite-volume cells for a lifting surface. The Navier-Stokes equations are then discretized over these cells using numerical methods. For an unsteady RANS calculation, the equations must also be discretized in time to capture the flow’s temporal evolution; a steady RANS calculation does not require physical-time resolution.
   

   

2. Apply the conservation equations (continuity and momentum) to each control volume. This involves discretizing the spatial and temporal derivatives in the equations. Typical schemes for solving the equations for the flow velocities and pressures include explicit and implicit time-stepping methods.
3. The resulting discretized equations generally form a system of nonlinear algebraic equations, which are linearized and solved iteratively. For an unsteady calculation, this iterative solution is performed at each physical time step. For a steady calculation, the equations are iterated directly, or advanced in pseudo-time, until the residuals and relevant flow quantities converge. Implicit methods solve a coupled system of equations during each iteration or time step, whereas explicit methods update the solution using previously known values.
4. Implement the selected turbulence model within the selected numerical framework. This step involves solving additional equations for turbulence quantities, such as those used in a turbulence closure model.
5. Apply appropriate boundary conditions to represent the problem’s physical boundaries. This may include specifying inlet, wall, and outlet conditions.
6. Iterate through the time steps until a steady state or a time-accurate solution is reached. Monitor the convergence criteria and adjust the solution until the changes between iterations are within acceptable tolerances or meet other specified criteria.
7. Analyze and post-process the numerical results to extract relevant information, including velocity profiles, pressure distributions, Reynolds stresses, TKE, and other turbulence quantities. The examples in Figure 13 illustrate RANS predictions of the wing-tip vortex roll-up using two different turbulence models. Although both solutions yield similar bulk flow properties, they differ quantitatively in terms of turbulence levels.

Numerical solvers for the RANS equations are available in various computational fluid dynamics (CFD) software packages. They often include preprocessing tools for mesh generation and turbulence model setup, as well as post-processing tools for visualizing and analyzing results. The choice of numerical method and solver depends on factors such as the complexity of the flow, available computational resources, and desired resolution and accuracy. While CFD predictions of the flow around an entire aircraft have become increasingly capable, as shown in Figure 14, their accuracy depends on grid resolution, geometry fidelity, turbulence modeling, boundary conditions, and validation against experimental or flight data. Confidence is usually highest for attached flows and lower for strongly separated, massively unsteady, or highly complex turbulent flows.

Remember that turbulence models used in CFD are empirical or semi-empirical closures, so their predictions must be interpreted within the assumptions and calibration range of the model. Solving the flow field numerically can reveal detailed mean-flow structures and modeled turbulence quantities, but these results should not be mistaken for a complete first-principles resolution of all turbulent scales. Nevertheless, the remarkable details of the flows around the complete airplane can now be computed. Ultimately, however, all CFD solutions must be considered tentative and validated against experiments or another benchmark.

Turbulence & Combustion

In many instances, turbulence is a desirable outcome in fluid dynamics, particularly in combustion processes. In piston, jet, and rocket engines, the rate of combustion is often limited by the rate at which the fuel and oxidizer can be mixed. Turbulence enhances this mixing process by increasing the transport of mass, momentum, and energy over a wide range of length scales.

For a simple one-dimensional example, the turbulent transport of a scalar quantity such as fuel concentration or temperature can be described in a manner analogous to momentum transport, i.e., using

(65)  

where represents a scalar quantity (such as species concentration or temperature) and is a measure of turbulent diffusivity. By analogy with eddy viscosity models, then

(66)  

showing that turbulent mixing is closely related to turbulent momentum transport. The relative importance of mixing and chemical reaction rates is often described using the Damköhler number, i.e.,

(67)  

When , mixing is slower than the chemical reactions, and the process is said to be mixing-limited. When , the reactions are slower, and the process is kinetics-limited. In most practical propulsion systems, combustion is primarily mixing-limited.

Turbulence enhances mixing by continuously deforming fluid elements and bringing regions of different composition into proximity. In premixed combustion, turbulence wrinkles and distorts the flame front, increasing its surface area and generally increasing the overall burning rate. Under some combustion regimes, turbulence may also broaden the flame structure. In non-premixed combustion, turbulence enhances mixing across the reaction zone, thereby increasing the rate of heat release. Figure 15 illustrates these processes for a turbulent fuel jet. As the fuel issues from the nozzle, it breaks up into droplets, evaporates, and mixes with the surrounding air. Turbulent motion entrains ambient fluid into the jet, supplying oxidizer to the fuel-rich region and establishing the conditions required for ignition.

In the near-field region, strong velocity gradients and rapid mixing lead to the formation of an ignition zone once flammable conditions are reached. Further downstream, a diffusion flame forms, where the rate of heat release is governed primarily by turbulent mixing. Engineers rely on both computational and experimental methods to model these interactions, enabling the design of combustion and propulsion systems that maximize performance while minimizing fuel consumption and emissions. The lower portion of Figure 15 shows a corresponding CFD simulation of a turbulent, reacting jet, in which the temperature contours reveal the flame structure. The broadened and irregular reaction zone reflects the influence of turbulent eddies over a range of length scales, while the spreading of the high-temperature region demonstrates the dominant role of turbulent mixing in determining the flame structure.

Reducing Turbulence

Existing turbulence cannot generally be eliminated instantaneously. Instead, turbulent kinetic energy decays as energy is transferred toward progressively smaller scales, where viscosity converts it into heat. Providing sufficient downstream distance or time for this decay is therefore an important part of many turbulence-reduction systems.

Turbulence may be reduced by weakening the disturbances and velocity gradients that generate it, delaying or preventing laminar-to-turbulent transition, removing large-scale swirl and flow nonuniformity, or allowing existing turbulent fluctuations to decay. The appropriate method depends on the flow and the engineering objective.

General Methods of Turbulence Reduction

For flows over surfaces, turbulence generation and boundary-layer transition can be reduced by using smooth contours, avoiding abrupt geometric changes, and minimizing surface roughness, gaps, waviness, and contamination. Favorable pressure gradients can also stabilize a boundary layer and delay the amplification of disturbances. In some applications, boundary-layer suction is used to remove low-momentum fluid near the surface and suppress instability growth.

In pipes, ducts, and other internal flows, gradual area changes, smooth walls, well-designed inlets, and the removal of swirl and upstream disturbances can reduce turbulence generation and flow nonuniformity. Reducing the Reynolds number, when practical, increases the relative influence of viscosity and may delay or suppress transition. Active flow-control methods, including suction, blowing, localized heating, plasma actuators, and feedback-controlled forcing, may also be used to weaken turbulent fluctuations or delay their development.

Reducing Turbulence in Wind Tunnels

Reductions in turbulence intensity can be achieved using turbulence screens, also known as anti-turbulence screens. These screens are made from fine-wire mesh with selected wire diameters and mesh spacings, as shown in Figure 16. They are usually positioned in the settling chamber upstream of the contraction to the test section. The screens disrupt large-scale velocity nonuniformities and break large turbulent structures into smaller ones that decay more rapidly downstream.

The decay of turbulence downstream of a grid or screen can be approximated by a power-law relation of the form

(68)  

where is the rms streamwise turbulence intensity normalized by the local mean velocity , is the downstream distance from the screen, is a characteristic mesh spacing, is an empirical decay constant, and is an empirical decay exponent. Values of are often of order unity but depend on the screen geometry, Reynolds number, solidity, and downstream distance. Equation 68 shows that reducing the characteristic turbulence length scale through successive screens can accelerate turbulence decay.

Figure 16 illustrates this mechanism. As the flow passes through each screen, large turbulent structures are disrupted by the mesh and replaced by smaller-scale disturbances. Because the characteristic turbulence length scale is reduced, viscous dissipation becomes more effective and the fluctuations decay more rapidly downstream. This process can be interpreted in terms of the turbulence energy cascade, as shown previously in Figure 9, in which energy is transferred from larger scales to smaller scales before being dissipated by viscosity.

At sufficiently small turbulent scales, an order-of-magnitude estimate for the viscous dissipation rate per unit mass is

(69)  

where is the characteristic velocity fluctuation associated with the scale . Therefore, transferring turbulent energy to smaller scales produces larger velocity gradients, allowing viscosity to dissipate the turbulent kinetic energy more effectively.Downstream of the final screen, the remaining small-scale fluctuations continue to decay within the settling chamber.

A settling chamber downstream of the final screen provides additional distance for the turbulence to decay and for the flow to become more spatially uniform. The flow then passes through a contraction section, where the mean velocity increases and the normalized turbulence intensity is further reduced. As a simple estimate,

(70)  

where is the contraction ratio. This relationship is an approximate scaling because the contraction also stretches and redistributes the turbulent fluctuations. The resulting flow entering the test section can have a very low turbulence intensity, although it is not laminar. Turbulence intensities below are commonly sought in low-turbulence aerodynamic wind tunnels.

The screens themselves can generate small-scale disturbances, depending partly on the Reynolds number based on the wire diameter. They also produce a pressure loss because the wires exert drag on the flow and cause local separation and mixing. The pressure loss across a screen is commonly expressed as

(71)  

where is the screen loss coefficient and

(72)  

is the local dynamic pressure at the screen. The value of depends primarily on the screen solidity, wire diameter, mesh geometry, and Reynolds number based on the wire diameter. A screen with greater solidity generally produces a greater reduction in large-scale flow nonuniformity but also causes a larger pressure loss.

When several screens are installed in succession, their pressure losses are cumulative, i.e.,

(73)  

In most wind tunnels, the screens are placed in the relatively large-area, low-speed settling chamber, where the local dynamic pressure is much lower than in the test section. This placement reduces the screen pressure losses when referenced to the test-section dynamic pressure. Nevertheless, excessive screen solidity or too many screens can significantly increase the required fan power and may generate additional small-scale disturbances. Consequently, a turbulence-reduction system must balance turbulence reduction, flow uniformity, pressure loss, and fan-power requirements.

Summary & Closure

A turbulent flow, by definition, is an unsteady flow because the fluid properties at a given point in the flow continuously change over time. This behavior contrasts with smooth, laminar flow, in which the fluid moves in layers with minimal mixing. Turbulent flows are more complex and challenging to analyze than laminar flows, requiring alternative mathematical models and computational methods. In aerospace engineering, turbulent flows are a fundamental phenomenon that significantly influences the performance of all flight vehicles. Random fluctuations in flow velocity and pressure characterize turbulence. Therefore, it poses a significant challenge for engineers to understand and model, as it might affect the performance and characteristics of flight vehicles. In particular, the intricate interplay among Reynolds stresses, vorticity generation, and flow separation requires a deeper understanding to mitigate their adverse effects.

Mathematical models, such as the Navier-Stokes equations, can be used to describe turbulent flows. However, solving these equations across all turbulence scales remains challenging. Engineers can employ advanced computational fluid dynamics (CFD) simulations in conjunction with wind-tunnel testing to understand and predict the effects of turbulence on the aerodynamics of flight vehicles. However, because of turbulence’s complex, apparently random nature, CFD simulations, such as RANS, and experiments must be conducted in tandem to study and analyze turbulent flows. It is essential to select an appropriate turbulence model that accounts for the specific flow characteristics and available computational resources. Different turbulence models have their strengths and limitations, and the choice depends on factors such as flow conditions, flow geometry, and desired accuracy. What is clear, however, is that continued understanding of turbulence’s complex characteristics is essential for optimizing future aircraft designs and improving fuel efficiency.