In turbulence models that employ the Boussinesq approach, the central issue is how the eddy viscosity is computed. The model proposed by Spalart and Allmaras  [ 26 ] solves a transport equation for a quantity that is a modified form of the turbulent kinematic viscosity.

The standard k-ε, RNG and realizable k-ε models have similar forms, with transport equations for k and ε. The major differences in the models are as follows:

This section describes the Reynolds-averaging method for calculating turbulent effects and provides an overview of the issues related to choosing an advanced turbulence model in Ansys Icepak. The transport equations, methods of calculating turbulent viscosity, and model constants are presented separately for each model. The features that are essentially common to both models follow, including turbulent production, generation due to buoyancy, and modeling heat transfer.

Reynolds (Ensemble) Averaging

The advanced turbulence models in Ansys Icepak are based on Reynolds averages of the governing equations. In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged) and fluctuating components. For the velocity components:

(40–16)

where and are the mean and instantaneous velocity components (i = 1, 2, 3).

Likewise, for pressure and other scalar quantities:

(40–17)

where φ denotes a scalar such as pressure or energy. Substituting expressions of this form for the flow variables into the instantaneous continuity and momentum equations and taking a time (or ensemble) average (and dropping the overbar on the mean velocity, ) yields the ensemble-averaged momentum equations. They can be written in Cartesian tensor form as:

(40–18)

(40–19)

(40–20)

Boussinesq Approach

The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation 40–20 be appropriately modeled. A common method employs the Boussinesq hypothesis [ 8 ] to relate the Reynolds stresses to the mean velocity gradients:

(40–21)

The Boussinesq hypothesis is used in the Spalart-Allmaras model and the k-ε models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, μ_t. In the case of the Spalart-Allmaras model, only one additional transport equation (representing turbulent viscosity) is solved. In the case of the k-ε models, two additional transport equations (for the turbulence kinetic energy, k, and the turbulence dissipation rate, ε) are solved, and μ_t is computed as a function of k and ε. The disadvantage of the Boussinesq hypothesis as presented is that it assumes μ_t is an isotropic scalar quantity, which is not strictly true.

Choosing an Advanced Turbulence Model

This section provides an overview of the issues related to the advanced turbulence models provided in Ansys Icepak.

The Spalart-Allmaras Model

The Spalart-Allmaras model is a relatively simple one-equation model that solves a modeled transport equation for the kinematic eddy (turbulent) viscosity. This embodies a relatively new class of one-equation models in which it is not necessary to calculate a length scale related to the local shear layer thickness. The Spalart-Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity for turbomachinery applications.

On a cautionary note, however, the Spalart-Allmaras model is still relatively new, and no claim is made regarding its suitability to all types of complex engineering flows. For instance, it cannot be relied on to predict the decay of homogeneous, isotropic turbulence. Furthermore, one-equation models are often criticized for their inability to rapidly accommodate changes in length scale, such as might be necessary when the flow changes abruptly from a wall-bounded to a free shear flow.

The Standard k-ε Model

The simplest "complete models" of turbulence are two-equation models in which the solution of two separate transport equations allows the turbulent velocity and length scales to be independently determined. The standard k-ε model in Ansys Icepak falls within this class of turbulence model and has become the workhorse of practical engineering flow calculations in the time since it was proposed by Launder and Spalding  [ 18 ]. Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. It is a semi-empirical model, and the derivation of the model equations relies on phenomenological considerations and empiricism.

As the strengths and weaknesses of the standard k-ε   model have become known, improvements have been made to the model to improve its performance. One of these variants is available in Ansys Icepak: the RNG k-ε model [ 29 ].

The RNG k-ε Model

The RNG k-ε model was derived using a rigorous statistical technique (called renormalization group theory). It is similar in form to the standard k-ε model, but includes the following refinements:

These features make the RNG k-ε  model more accurate and reliable for a wider class of flows than the standard k-ε  model.

The Realizable k-ε Model

The realizable k-ε  model is a relatively recent development and differs from the standard k-ε  model in two important ways:

The term "realizable" means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard k-ε  model nor the RNG k-ε  model is realizable.

An immediate benefit of the realizable k-ε  model is that it more accurately predicts the spreading rate of both planar and round jets. It is also likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation.

The Enhanced Two-Equation Models

The k-ε  models are primarily valid for turbulent core flows (that is, the flow in the regions somewhat far from walls). Consideration therefore needs to be given as to how to make these models suitable for wall-bounded flows.

Turbulent flows are significantly affected by the presence of walls. Obviously, the mean velocity field is affected through the no-slip condition that has to be satisfied at the wall. However, the turbulence is also changed by the presence of the wall in non-trivial ways. Very close to the wall, viscous damping reduces the tangential velocity fluctuations, while kinematic blocking reduces the normal fluctuations. Toward the outer part of the near-wall region, however, the turbulence is rapidly augmented by the production of turbulence kinetic energy due to the large gradients in mean velocity.

The near-wall modeling significantly impacts the fidelity of numerical solutions, inasmuch as walls are the main source of mean vorticity and turbulence. It is in the near-wall region where the solution variables have large gradients and where the momentum and other scalar transports are the greatest. Therefore, accurate representation of the flow in the near-wall region determines successful predictions of wall-bounded turbulent flows.

Numerous experiments have shown that the near-wall region can be largely subdivided into three layers. In the innermost layer, called the "viscous sublayer", the flow is almost laminar, and the (molecular) viscosity plays a dominant role in momentum and heat or mass transfer. In the outer layer, called the fully-turbulent layer, turbulence plays a major role. Finally, there is an interim region between the viscous sublayer and the fully turbulent layer where the effects of molecular viscosity and turbulence are equally important.

To more accurately resolve the flow near the wall, the enhanced two-equation models combine three k-ε  models (standard, RNG and realizable) with enhanced wall treatment.

Enhanced Wall Treatment

Enhanced wall treatment is a near-wall modeling method that combines a two-layer model with enhanced wall functions  [ 3 9 14 15 27 28 ].

In the two-layer model, the viscosity-affected near-wall region is completely resolved all the way to the viscous sublayer. The two-layer approach is an integral part of the enhanced wall treatment and is used to specify both ε and the turbulent viscosity in the near-wall cells. In this approach, the whole domain is subdivided into a viscosity-affected region and a fully-turbulent region. The demarcation of the two regions is determined by a wall-distance-based, turbulent Reynolds number.

If the near-wall mesh is fine enough to be able to resolve the laminar sublayer (typically ), then the enhanced wall treatment will be identical to the traditional two-layer zonal model. However, the restriction that the near-wall mesh must be sufficiently fine everywhere might impose too large a computational requirement. Ideally, then, one would like to have a near-wall formulation that can be used with coarse meshes (usually referred to as wall-function meshes) as well as fine meshes (low-Reynolds-number meshes). In addition, excessive error should not be incurred for intermediate meshes that are too fine for the near-wall cell centroid to lie in the fully turbulent region, but also too coarse to properly resolve the sublayer.

To achieve the goal of having a near-wall modeling approach that will possess the accuracy of the standard two-layer approach for fine near-wall meshes and will not significantly reduce accuracy for wall-function meshes, Ansys Icepak combines the two-layer model with enhanced wall functions to result in the enhanced wall treatment.

Computational Effort: CPU Time and Solution Behavior

The standard k-ε  model clearly requires more computational effort than the Spalart-Allmaras model since an additional transport equation is solved. The realizable k-ε  model requires only slightly more computational effort than the standard k-ε  model. However, due to the extra terms and functions in the governing equations and a greater degree of nonlinearity, computations with the RNG k-ε  model tend to take 10-15% more CPU time than with the standard k-ε  model.

Aside from the time per iteration, the choice of turbulence model can affect the ability of Ansys Icepak to obtain a converged solution. For example, the standard k-ε  model is known to be slightly over-diffusive in certain situations, while the RNG k-ε  model is designed such that the turbulent viscosity is reduced in response to high rates of strain. Because diffusion has a stabilizing effect on the numerics, the RNG model is more likely to be susceptible to instability in steady-state solutions. However, this should not necessarily be seen as a disadvantage of the RNG model, since these characteristics make it more responsive to important physical instabilities such as time-dependent turbulent vortex shedding.

The Spalart-Allmaras Model

In its original form, the Spalart-Allmaras model is effectively a low-Reynolds-number model, requiring the viscous-affected region of the boundary layer to be properly resolved. In Ansys Icepak, however, the Spalart-Allmaras model has been implemented to use wall functions when the mesh resolution is not sufficiently fine. This might make it the best choice for relatively crude simulations on coarse meshes where accurate turbulent flow computations are not critical. Furthermore, the near-wall gradients of the transported variable in the model are much smaller than the gradients of the transported variables in the k-ε  models. This might make the model less sensitive to numerical error when non-layered meshes are used near walls.

Transport Equation for the Spalart-Allmaras Model

The transported variable in the Spalart-Allmaras model, , is identical to the turbulent kinematic viscosity except in the near-wall (viscous-affected) region. The transport equation for is

(40–22)

where G^ν is the production of turbulent viscosity and Y_ν is the destruction of turbulent viscosity that occurs in the near-wall region due to wall blocking and viscous damping. and C_b2 are constants and ν is the molecular kinematic viscosity. is a user-defined source term. Note that since the turbulence kinetic energy k is not calculated in the Spalart-Allmaras model, the last term in Equation 40–21 is ignored when estimating the Reynolds stresses.

Modeling the Turbulent Viscosity

The turbulent viscosity, μ_t , is computed from

(40–23)

where the viscous damping function, f_v1 , is given by

(40–24)

and

(40–25)

Modeling the Turbulent Production

The production term, G_ν , is modeled as

(40–26)

where

(40–27)

and

(40–28)

C_b1 and κ are constants, d is the distance from the wall, and S is a scalar measure of the deformation tensor. By default in Ansys Icepak, as in the original model proposed by Spalart and Allmaras, S is based on the magnitude of the vorticity:

(40–29)

where Ω_ij is the mean rate-of-rotation tensor and is defined by

(40–30)

The justification for the default expression for S is that, for the wall-bounded flows that were of most interest when the model was formulated, turbulence is found only where vorticity is generated near walls. However, it has since been acknowledged that one should also take into account the effect of mean strain on the turbulence production, and a modification to the model has been proposed  [[66]] and incorporated into Ansys Icepak.

This modification combines measures of both rotation and strain tensors in the definition of S:

(40–31)

where

(40–32)

with the mean strain rate, S_ij , defined as

(40–33)

Including both the rotation and strain tensors reduces the production of eddy viscosity and consequently reduces the eddy viscosity itself in regions where the measure of vorticity exceeds that of strain rate. One such example can be found in vortical flows, that is, flow near the core of a vortex subjected to a pure rotation where turbulence is known to be suppressed. Including both the rotation and strain tensors more correctly accounts for the effects of rotation on turbulence. The default option (including the rotation tensor only) tends to overpredict the production of eddy viscosity and hence overpredicts the eddy viscosity itself in certain circumstances.

Modeling the Turbulent Destruction

The destruction term is modeled as

(40–34)

where

(40–35)

(40–36)

(40–37)

C_w1 , C_w2 , and C_w3 are constants, and is given by Equation 40–27. Note that the modification described above to include the effects of mean strain on S will also affect the value of used to compute r.

Model Constants

The model constants , and have the following default values  [ 26 ]:

(40–38)

(40–39)

Wall Boundary Conditions

At walls, the modified turbulent kinematic viscosity, , is set to zero.

When the mesh is fine enough to resolve the laminar sublayer, the wall shear stress is obtained from the laminar stress-strain relationship:

(40–40)

If the mesh is too coarse to resolve the laminar sublayer, it is assumed that the centroid of the wall-adjacent cell falls within the logarithmic region of the boundary layer, and the law-of-the-wall is employed:

(40–41)

where u is the velocity parallel to the wall, u_τ is the shear velocity, y is the distance from the wall, κ is the von Kármán constant (0.4187), and E = 9.793.

Convective Heat and Mass Transfer Modeling

In Ansys Icepak, turbulent heat transport is modeled using the concept of Reynolds’ analogy to turbulent momentum transfer. The "modeled" energy equation is thus given by the following:

(40–42)

where k, in this case, is the thermal conductivity, E is the total energy, and (τ_ij)_eff is the deviatoric stress tensor, defined as

(40–43)

The term involving (τ_ij)_eff represents the viscous heating. The default value of the turbulent Prandtl number is 0.85. Turbulent mass transfer is treated similarly, with a default turbulent Schmidt number of 0.7.

Wall boundary conditions for scalar transport are handled analogously to momentum, using the appropriate "law-of-the-wall".

Two-Equation (Standard k-ε) Turbulence Model

The two-equation turbulence model (also known as the standard k-ε model) is more complex than the zero-equation model. The standard k-ε model  [ 18 ] is a semi-empirical model based on model transport equations for the turbulent kinetic energy (k) and its dissipation rate (ε). The model transport equation for k is derived from the exact equation, while the model transport equation for ε is obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart.

In the derivation of the standard k-ε  model, it is assumed that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k-ε model is therefore valid only for fully turbulent flows.

Transport Equations for the Standard k-ε  Model

The turbulent kinetic energy,k, and its rate of dissipation, ε, are obtained from the following transport equations:

(40–44)

and

(40–45)

In these equations, G_k represents the generation of turbulent kinetic energy due to the mean velocity gradients, calculated as described later in this section. G_b is the generation of turbulent kinetic energy due to buoyancy, calculated as described later in this section. C_1ε ,C_2ε , and C_3ε are constants. σ_k and σ_ε are the turbulent Prandtl numbers for k and ε, respectively.

Modeling the Turbulent Viscosity

The "eddy" or turbulent viscosity , μ_t , is computed by combining k and ε as follows:

(40–46)

where C_μ is a constant.

Model Constants

The model constants C_1ε , C_2ε . C_3μ , σ_k , and σ_ε have the following default values  [ 18 ]:

(40–47)

These default values have been determined from experiments with air and water for fundamental turbulent shear flows including homogeneous shear flows and decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows.

The RNG k-ε  Model

The RNG-based k-ε turbulence model is derived from the instantaneous Navier-Stokes equations, using a mathematical technique called "renormalization group" (RNG) methods. The analytical derivation results in a model with constants different from those in the standard k-ε  model, and additional terms and functions in the transport equations for k and ε. A more comprehensive description of RNG theory and its application to turbulence can be found in [ 4 ].

Transport Equations for the RNG ε  Model

The RNG k-ε model has a similar form to the standard ε  model:

(40–48)

and

(40–49)

In these equations, G_k represents the generation of turbulent kinetic energy due to the mean velocity gradients, calculated as described later in this section. G_b is the generation of turbulent kinetic energy due to buoyancy, calculated as described later in this section. The quantities α _k and α_ε are the inverse effective Prandtl numbers for k and ε, respectively.

Modeling the Effective Viscosity

The scale elimination procedure in RNG theory results in a differential equation for turbulent viscosity:

(40–50)

where

(40–51)

Equation 40–50 is integrated to obtain an accurate description of how the effective turbulent transport varies with the effective Reynolds number (or eddy scale), allowing the model to better handle low-Reynolds-number and near-wall flows.

In the high-Reynolds-number limit, Equation 40–50 gives

(40–52)

with C_μ = 0.0845, derived using RNG theory. It is interesting to note that this value of _Cμ is very close to the empirically-determined value of 0.09 used in the standard k-ε  model.

In Ansys Icepak, the effective viscosity is computed using the high-Reynolds-number form in Equation 40–52.

Calculating the Inverse Effective Prandtl Numbers

The inverse effective Prandtl numbers α _k and α _ε are computed using the following formula derived analytically by the RNG theory:

(40–53)

where α ₀ = 1.0. In the high-Reynolds-number limit (μ_mol ∕μ_eff ≪1), α _k = α _ε ≈ 1.393.

The R_ε Term in the ε Equation

The main difference between the RNG and standard k-ε models lies in the additional term in the ε equation given by

(40–54)

where η≡Sk ∕ ε, η₀ = 4.38, β = 0.012.

The effects of this term in the RNG ε equation can be seen more clearly by rearranging Equation 40–49. Using Equation 40–54, the last two terms in Equation 40–49 can be merged, and the resulting ε equation can be rewritten as

(40–55)

where C_2ε ^∗ is given by

(40–56)

In regions where η<<η₀ , the R term makes a positive contribution, and C_2ε ^∗ becomes larger than C_2ε . In the logarithmic layer, for instance, it can be shown that η≈3.0, giving C_2ε ^∗ ≈ 2.0, which is close in magnitude to the value of C_2ε in the standard k-ε model (1.92). As a result, for weakly to moderately strained flows, the RNG model tends to give results largely comparable to the standard k-ε model.

In regions of large strain rate (η η₀ ), however, the R term makes a negative contribution, making the value of C_2ε ^∗ less than C_2ε . In comparison with the standard k-ε model, the smaller destruction of ε augments ε, reducing k and eventually the effective viscosity. As a result, in rapidly strained flows, the RNG model yields a lower turbulent viscosity than the standard k-ε model.

Thus, the RNG model is more responsive to the effects of rapid strain and streamline curvature than the standard k-ε  model, which explains the superior performance of the RNG model for certain classes of flows.

Model Constants

The model constants C_1ε and C_2ε in Equation 40–49 have values derived analytically by the RNG theory. These values, used by default in Ansys Icepak, are C_1ε = 1.42, C_2ε = 1.68.

Modeling Turbulent Production in the k-ε Models

From the exact equation for the transport of k, the term G_k , representing the production of turbulent kinetic energy, can be defined as

(40–57)

To evaluate G_k in a manner consistent with the Boussinesq hypothesis,

(40–58)

where S is the modulus of the mean rate-of-strain tensor, defined as

(40–59)

with the mean strain rate S_ij given by

(40–60)

Effects of Buoyancy on Turbulence in the k-ε Models

When a non-zero gravity field and temperature gradient are present simultaneously, the k-ε models in Ansys Icepak account for the generation of k due to buoyancy (G_b in Equations  Equation 40–44 and Equation 40–48), and the corresponding contribution to the production of ε in Equations  Equation 40–45 and Equation 40–49.

The generation of turbulence due to buoyancy is given by

(40–61)

where Pr_t is the turbulent Prandtl number for energy. For the standard k-ε model, the default value of Pr_t is 0.85. In the case of the RNG k-ε model, Pr_t = 1∕α, where α is given by Equation 40–53, but with α ₀ = 1∕Pr = k∕μc_p. The coefficient of thermal expansion, β, is defined as

(40–62)

It can be seen from the transport equation for k (Equation 40–44 or Equation 40–48) that turbulent kinetic energy tends to be augmented (G_b 0) in unstable stratification. For stable stratification, buoyancy tends to suppress the turbulence (G_b <0). In Ansys Icepak, the effects of buoyancy on the generation of k are always included when you have both a non-zero gravity field and a non-zero temperature (or density) gradient.

While the buoyancy effects on the generation of k are relatively well understood, the effect on ε is less clear. In Ansys Icepak, by default, the buoyancy effects on ε are neglected simply by setting G_b to zero in the transport equation for ε (Equation 40–45 or Equation 40–49).

The degree to which ε is affected by the buoyancy is determined by the constant C_3ε . In Ansys Icepak, C_3ε is not specified, but is instead calculated according to the following relation  [ 7 ]:

(40–63)

where v is the component of the flow velocity parallel to the gravitational vector and u is the component of the flow velocity perpendicular to the gravitational vector. In this way, C_3ε will become 1 for buoyant shear layers for which the main flow direction is aligned with the direction of gravity. For buoyant shear layers that are perpendicular to the gravitational vector, _3ε will become zero.

Convective Heat Transfer Modeling in the k-ε Models

In Ansys Icepak, turbulent heat transport is modeled using the concept of Reynolds’ analogy to turbulent momentum transfer. The "modeled" energy equation is thus given by the following:

(40–64)

where E is the total energy and k_eff is the effective conductivity.

For the standard k-ε model,k_eff is given by

(40–65)

with the default value of the turbulent Prandtl number set to 0.85.

For the RNG k-ε model, the effective thermal conductivity is

(40–66)

where α is calculated from Equation 40–53, but with α ₀ = 1∕Pr = k∕μc_p.

The fact that α varies with μ_mol ∕μ_eff , as in Equation 40–53, is an advantage of the RNG k-ε model. It is consistent with experimental evidence indicating that the turbulent Prandtl number varies with the molecular Prandtl number and turbulence  [ 17 ]. Equation 40–53 works well across a very broad range of molecular Prandtl numbers, from liquid metals (Pr≈10⁻) to paraffin oils (Pr≈10³), which allows heat transfer to be calculated in low-Reynolds-number regions. Equation 40–53 smoothly predicts the variation of effective Prandtl number from the molecular value (α = ∕Pr) in the viscosity-dominated region to the fully turbulent value (α = 1.393) in the fully turbulent regions of the flow.

k-ω SST Model

The k-ω SST model is based on the coupling of the SST k-ω transport equations, one for the intermittency and one for the transition onset criteria, in terms of momentum-thickness Reynolds number. An Ansys empirical correlation (Langtry and Menter) has been developed to cover standard bypass transition as well as flows in low-free-stream turbulence environments. The k-ω SST model can be used as a low Reynolds number turbulence model. It also exhibits good behavior in adverse pressure gradients and separating flow.

Transport Equations for the Transition SST Model

The transport equation for the intermittency is defined as:

(40–67)

The transition sources are defined as follows:

(40–68)

where is the strain rate magnitude, is an empirical correlation that controls the length of the transition region, and and hold the values of 2 and 1, respectively. The destruction/relaminarization sources are defined as follows:

(40–69)

where is the vorticity magnitude. The transition onset is controlled by the following functions:

(40–70)

(40–71)

(40–72)

is the critical Reynolds number where the intermittency first starts to increase in the boundary layer. This occurs upstream of the transition Reynolds number and the difference between the two must be obtained from an empirical correlation. Both the and correlations are functions of .

The constants for the intermittency equation are:

(40–73)

Separation-Induced Transition Correction

The modification for separation-induced transition is:

(40–74)

Here, is a constant with a value of 2.

The model constants in Equation 40–74 have been adjusted from those of Menter et al. [30] in order to improve the predictions of separated flow transition. The main difference is that the constant that controls the relation between and was changed from 2.193, its value for a Blasius boundary layer, to 3.235, the value at a separation point where the shape factor is 3.5  [30]. The boundary condition for at a wall is zero normal flux, while for an inlet, is equal to 1.0.

The transport equation for the transition momentum thickness Reynolds number is

(40–75)

The source term is defined as follows:

(40–76)

(40–77)

(40–78)

(40–79)

The model constants for the equation are:

(40–80)

The boundary condition for at a wall is zero flux. The boundary condition for at an inlet should be calculated from the empirical correlation based on the inlet turbulence intensity.

The model contains three empirical correlations. is the transition onset as observed in experiments. This has been modified from Menter et al.  [30] in order to improve the predictions for natural transition. It is used in Equation 40–75. is the length of the transition zone and is substituted in Equation 40–67. is the point where the model is activated in order to match both and , and is used in Equation 40–71. At present, these empirical correlations are proprietary and are not given in this User’s guide.

(40–81)

The first empirical correlation is a function of the local turbulence intensity, , and the Thwaites’ pressure gradient coefficient is defined as

(40–82)

where is the acceleration in the streamwise direction.

Coupling the Transition Model and SST Transport Equations

The transition model interacts with the SST turbulence model by modification of the -equation, as follows:

(40–83)

(40–84)

(40–85)

where and are the original production and destruction terms for the SST model. Note that the production term in the -equation is not modified. The rationale behind the above model formulation is given in detail in Menter et al.  [30].

In order to capture the laminar and transitional boundary layers correctly, the mesh must have a of approximately one. If the is too large (that is, > 5), then the transition onset location moves upstream with increasing . It is recommended that you use the bounded second order upwind based discretization for the mean flow, turbulence and transition equations.

Specifying Inlet Turbulence Levels

It has been observed that the turbulence intensity specified at an inlet can decay quite rapidly depending on the inlet viscosity ratio () (and hence turbulence eddy frequency). As a result, the local turbulence intensity downstream of the inlet can be much smaller than the inlet value (see Figure 40.1: Exemplary Decay of Turbulence Intensity (Tu) as a Function of Streamwise Distance (x)). Typically, the larger the inlet viscosity ratio, the smaller the turbulent decay rate. However, if too large a viscosity ratio is specified (that is, > 100), the skin friction can deviate significantly from the laminar value. There is experimental evidence that suggests that this effect occurs physically; however, at this point it is not clear how accurately the transition model reproduces this behavior. For this reason, if possible, it is desirable to have a relatively low (that is, 1 – 10) inlet viscosity ratio and to estimate the inlet value of turbulence intensity such that at the leading edge of the blade/airfoil, the turbulence intensity has decayed to the desired value. The decay of turbulent kinetic energy can be calculated with the following analytical solution:

(40–86)

For the SST turbulence model in the freestream the constants are:

(40–87)

The time scale can be determined as follows:

(40–88)

where is the streamwise distance downstream of the inlet and is the mean convective velocity. The eddy viscosity is defined as:

(40–89)

The decay of turbulent kinetic energy equation can be rewritten in terms of inlet turbulence intensity () and inlet eddy viscosity ratio () as follows:

(40–90)

Figure 40.1: Exemplary Decay of Turbulence Intensity (Tu) as a Function of Streamwise Distance (x)