On Boundary-Value Problems for RANS Equations and Two-Equation Turbulence Models

In this paper we consider the Reynolds-averaged Navier–Stokes (RANS) equations. The RANS equations are based upon Reynolds expansions for the dependent variables and time (Reynolds) averaging. In practice, this definition is extended to include applying a combination of Reynolds expansions (for the pressure and density) and mass weighted (Favre) expansions for the other dependent flow variables (see for example the textbooks of Wilcox and Pope [1, 2]). In so doing, the time-averaged equations can retain the same form as their laminar flow counterparts. The resulting mean flow momentum equations are simply augmented by an additional stress term (a Reynolds stress), and the mean flow energy equation includes an additional heat flux term (Reynolds heat flux) and work term due to the Reynolds stresses. To close the system of equations, some form of modeling is required to determine the Reynolds stresses and Reynolds heat fluxes.

Currently, in engineering applications, partial differential or integral equations of a transport type are generally used in modeling the effects of turbulence (see [3‐9]). One approach for modeling the additional terms appearing in the RANS equations, such as Reynolds stresses, is to invoke the eddy viscosity hypothesis (the Boussinesq approximation) in which the Reynolds stress tensor is proportional to the mean strain rate tensor. This constitutive relation requires the determination of a viscosity coefficient that is called an eddy viscosity. The eddy viscosity depends on two scales: a velocity scale and a length scale. These scales can be determined using transport equations. Usually, turbulence modeling formulations use one or two equations to determine the scales. For one-equation models, the velocity scale comes from solving for either the turbulence kinetic energy k or a quantity that can be related to k. The length scale for these models comes from an algebraic method that depends on physical location and whether the flow is a wall-bounded shear layer or a free shear layer. Two-equation models are often considered since the two scales can be computed from the two transport equations, one for k and the other for a quantity related to the length scale, such as the dissipation rate of k. Two advantages in using one-equation or two-equation models are that they avoid the necessity of solving directly for the Reynolds stresses and simplify considerably the modeling and solution requirements (e.g., typical Reynolds stress models require solving six equations for the Reynolds stresses and one equation for the length scale). The focus in this paper concerns two-equation models.

In the development, transformation and improvement of turbulence models the main focus is the variation of the equations themselves. This approach is interesting from the point of view that the main drivers for solutions are the boundary values. Roughly speaking, the predictions using the model cannot become better than the boundary values enforced. To understand possible shortcomings and behavior of the RANS equations in combination with turbulence models, the impact and influence of the formulation of the boundary conditions cannot be overstated. Both the complete boundary-value problem and the influence of the boundary values on the solution need to be understood. For example, the boundary values need to be chosen in such a way that it is at least possible to compute a solution. When one does not know in which way to choose the boundary values correctly, that is the user needs to make a guess, one cannot in general expect to find a solution of the corresponding boundary-value problem.

Against the background of this approach the following questions ariseThe principal objective of this paper is the presentation and discussion of the complete boundary-value problem for the RANS equations coupled with two transport equations for turbulence modeling. In this paper, the meaning of a well-defined boundary-value problem is one in which the equations and boundary conditions are defined, as well as the near-field and far-field behavior of the solution variables. Moreover, the consistency requirements between the specified values of the dependent variables and their decay behavior in the interior of the flow field are addressed in detail. As part of this overall objective, it is a goal to draw attention to which way boundary conditions already establish properties of solutions to the equations. An additional subject pertaining to this system of PDEs is the effect of transforming the transport equations in an effort to remove the singular behavior of the dissipation (or dissipation rate) at a solid surface boundary. In solving any system of PDEs it is critical to establish some understanding of well-posedness. This subject is also considered. An example is given that gives a strong indication of the ill-posedness of the problem governed by the transport equations of the turbulence model.

The question of solvability for the complete boundary-value problem yields directly to important mathematical questions. Namely, it is essential to return to the fundamental issue of constructing a well-posed problem. This issue is existential in solving both the Navier–Stokes and turbulence modeling equations. As pointed out previously, we cannot in general establish a well-posed problem for even the Navier–Stokes equations, in the sense of Hadamard [10] and supplemented with the condition of regularity. This mathematical issue already sets the stage for the issue of an ill-posed problem for the system of transport equations. Such a consequence exposes the probability of anomalous results or multiple solutions. At this point, there has been very little documented concerning the existence of multiple solutions when considering various turbulence models. Moreover, there is currently no systematic way of demonstrating multiple discrete solutions, as well as how many such solutions may exist, when modeling turbulence.

It is important to recognize that there are examples of multiple solutions in fluid mechanics, which have occurred when solving the transonic potential flow equation [11] and the Euler equations [12, 13]. There is also the discussion of Temam [14] concerning existence, uniqueness and regularity for the Navier–Stokes equations. In fact, Temam shows an example of non-uniqueness of solutions for the Navier–Stokes equations (i.e., regarding laminar flows) in which he provides proof based on topological arguments.

More recently, in 2014, Kamenetskiy et al. [15] presented numerical evidence of multiple solutions to the RANS equations for turbulent flows. They consider turbulent flow over an extruded two-dimensional (2-D) airfoil geometry and a trap wing configuration from the first AIAA High Lift Workshop [16]. Computations were performed for both the one-equation model of Spalart-Allmaras and the two-equation model of Wilcox. Three primary approaches were considered to find multiple solutions, which are as follows: Varying initial conditions, applying different techniques for enforcement of solid surface boundary conditions, using variations of implicit residual smoothing. For the trap wing case, the number of multiple solutions discovered with the methods depended on the mesh density. The meshes used (designated coarse, medium and fine) were from the AIAA High Lift Workshop. It should be pointed out that in the work of Kamenetskiy et al. numerical results are only classified as solutions when the residuals of the equations have been reduced to machine zero. As discussed previously, reduction of the residual to machine zero is essential. Otherwise, results in which the residual has only been reduced four or five orders of magnitude are affected by solver integration errors that can play a significant role in the behavior of the result, possibly making it quite different from the actual solution. Kamenetskiy et al. call such results pseudo-solutions. It is interesting that on a coarse mesh eight solutions were found, while only two solutions were discovered on the fine mesh. This suggests a connection to spatial resolution. Thus, the question immediately arises as to how many multiple solutions, if any, will be obtained when all the relevant scales of the turbulent flow are resolved. With a well resolved flow field (on a sufficiently fine mesh), it is distinctly possible that the number of multiple solutions is much greater than two solutions, which is a consequence of only small variations in the solutions. A final point is that Kamenetskiy et al. observed that the appearance of multiple solutions seems to be closely related to smooth body flow separation that occurs for flows over high-lift configurations. Here, we note that such occurrences of non-unique solutions may only be one member of the set of possible turbulent flows which have multiple solutions.

The authors have encountered both non-unique solutions as well as anomalous (nonphysical) results for the particular case of circulation control airfoil flows [17, 18]. Being aware of the fact that the forward problem, that is to find a solution of the boundary-value problem, is possibly ill-posed, we may ask the question vice versa: Is turbulence modeling its self ill-posed? And if so, what kinds of consequences follow from such a conclusion.

The article is organized as follows. Section 2 presents the governing equations of fluid flow, and Sect. 3 considers the additional equations required to model the effects of turbulence. Boundary conditions together with the corresponding boundary-value problems are discussed in Sect. 4. Proposed methods in the literature to deal with problems arising from the boundary conditions of the turbulent flow equations are introduced in Sect. 5. An interpretation of turbulence modeling as an inverse problem is given in Sect. 6. Numerical examples supporting the assertions made in this article are presented in Sect. 7.

To describe flow behavior we consider for the domain \(D\subset {\mathbb {R}}^m\), \(m=2,3,\) i.e., an open and connected set, and an interval \([0,T )\subset {\mathbb {R}}\), \(T>0\), the RANS equations in conservative form expressed in integral form by where the integral operators \(V_{D}\) and \(R_{\partial D}\) are given by and \(W: D\times [0,T)\rightarrow {\mathbb {R}}^{m+2}\),denotes the vector field of conserved variables and n is the unit outward normal on \(\partial D\). The terms \(f_c\) and \(f_v\) describe the convective and viscous contributionsHere \(e_i\) is the ith unit vector. The quantities \(\rho \), \(u=\left( u_1,\ldots ,u_m\right) ^T\), \(E\) andare the density, the velocity, the specific total energy, and the enthalpy of the fluid. The equation of statedefines the pressure \(p\), and \(\gamma \) is the gas dependent ratio of specific heats, which is given by 1.4 for air. Assuming that an effective viscosityis given and, using Stoke’s hypothesis, that the bulk viscosity satisfies \(\lambda = -2/3\mu _{\mathrm{eff}}\), the viscous stress tensor \(\tau =\tau \left( W\right) = \tau \left( W\left( x,t\right) \right) \) is given byand \({\mathcal {S}}\) denotes the strain rate, which is given by the symmetric part of the total derivative of flow velocity vector u,Throughout the paper \(\mathrm Id(x) = x\) denotes the identity operator. The missing viscous flux term for the energy equation is given by The effective viscosity \(\mu _{\mathrm{eff}}\) and effective conductivity \(\kappa _{\mathrm{eff}}\) are computed byand the laminar viscosity is given by Sutherland’s lawwhereby \(\rho _{\infty } > 0\) and \(u_{\infty } > 0\) denote some constant reference density and velocity, \(L>0\) is some constant reference length scale, \(Re>0\) is the corresponding Reynolds number,is Sutherland’s constant, \(\mathfrak {R}\) is the universal gas constant and the laminar Prandtl number is given by \(Pr_l :=0.72\).

In this article, we restrict ourselves to linear turbulence models represented by differential or integral equations. The solutions of these equations reveal additional quantities in the considered fluid. These occurring variables extend the degrees of freedom given by the conservative variables \(W\) by a further unknown functionHere \(N_{t}\in {\mathbb {N}}\) depends on the turbulence model. In this report we haveSince we mainly deal with two-equation turbulence models, one can assume \(N_t=2\) throughout this report. The Spalart-Allmaras model is mentioned here for completeness since results used for comparisons are presented in Sect. 7. Note, when for example using an algebraic model \(N_t = 0\), no additional unknowns need to be considered.

The additional variables required for the turbulence model are then used to to determine the eddy viscosity,which is required for (8). Given the eddy viscosity \(\mu _t\) the turbulent thermal conductivity is described by the algebraic relationFormula (12) is also required for (8). The determination of \(\mu _t\) closes the system of the Reynolds-averaged Navier–Stokes equations (1a).

In this section, we introduce the two-equation \(k\text {-}\omega \) models of interest, which are often used for aerodynamic applications. These are the \(k\text {-}\omega \) model of Wilcox introduced 1988 and the Shear Stress Transport  (SST) model by Menter [19]. Both models include differential equations for the unknown, positive functionswhere k is turbulence kinetic energy per unit mass and \(\omega \) the dissipation rate of turbulence kinetic energy. Throughout the literature two-equation turbulence models are typically presented in their differential form, though in finite-element and finite-volume codes an integral form is used.

Before we state the two transport equations for k and \(\omega \), we define \(t=\left( t_{ij}\right) _{1\le i, j\le m}\) and the Reynolds stress tensor \(\tau ^{\mathrm{RS}} = \left( \tau ^{\mathrm{RS}}_{ij}\right) _{1\le i, j\le m}\), using the strain rate tensors \({\mathcal {S}}\) and \(\overline{{\mathcal {S}}}\), given in (6) and (5), and the mean-molecular-stress sensor, \(t=\left( t_{ij}\right) _{1\le i, j\le m}\). Then,Additionally, according to \({\mathcal {S}}\) we define the vorticity \(\Omega \) as skew-symmetric part of the total derivative of flow velocity u,

So far, we have only stated the equations of interest. That is, the mean flow equations (1) together with a system of equations to determine the required eddy viscosity \(\mu _t\), for example (16) or (19). Naturally, for a closed representation, we need to formulate a corresponding boundary-value problem.

When solving two-equation models such as the \(k\text {-}\omega \) type or \(k\)-\(\varepsilon \) type, one typically encounters the problem of ensuring that turbulence variables remain positive during the iterations (see (13)). In general, negative values of k and / or \(\omega \) directly yield a breakdown of the iteration resulting in ”Not a number”. To avoid such problems, it was suggested in [25] to substitute variables k and \(\varepsilon \) or \(\omega \) byIn particular, the sole substitution for \(\omega \) found its way into several implementations, most often in the background of discontinuous Galerkin methods. Here, the idea has been picked up originally by Bassi et al. [26] and reused in [27]. Recently, a further application of such substitution was used in [28].

Naturally, substitutions (60) only make sense for nondimensional k, \(\varepsilon \) and \(\omega \). This is not a severe restriction, but depending on the implementation care must be taken. For example, typically implementations want to support flexible restart options. Then, knowledge about the output variable such as its kind of nondimensionalization and scaling is necessary to convert the variable.

Further substitutions discussed in the literature are (see [29‐31])These substitutions were in particular introduced to deal with the singular behavior of \(\omega \) indicated in (32).

To evaluate the formulation and the accuracy of a turbulence model is not an easy task. From our perspective the following four points are a minimum standard one has to consider: As soon as one of these criteria is not satisfied, certain doubts about the assertions made concerning a turbulence model arise. Unfortunately, throughout the literature about computational fluid dynamics, the explicit details of the computations are often hidden, and convincing arguments about convergence are also often missing. In particular, information about the actual formulation of boundary conditions as well as the possible impact of certain limitations of variables to stabilize the solution process is often missing. Even when all these criteria are satisfied and there is full evidence about the implementation, a strict conclusion about the accuracy of computed results is difficult to obtain. Typical validation measures are the comparison with measurements, which also come from a process which is inaccurate. This observation about this complex topic motivates one to possibly reconsider turbulence modeling as a parameter identification problem, typically an inverse problem, which is ill-posed.

To look at this point of view, we do not need to consider derivation of the RANS equations. Instead, we consider the system of equations (1a) such that it includes an unknown parameter \(\mu _t\ge 0\), which needs to be determined. For the simplest choice, which assumes \(\mu _t=0\), it is in general impossible to find steady-state solutions of (1a) for high Reynolds numbers. High Reynolds number computations using equations (1a) and \(\mu _t=0\) are called Direct Numerical Simulations (DNS), and the integral curves \(W_{\mathrm{lam}}=W_{\mathrm{lam}}(x,t)\) satisfying (1a) are in general unsteady.

There are many reasons why it is often not possible to numerically approximate \(W_{\mathrm{lam}}\) directly. Besides the mathematically open problem to prescribe meaningful physical initial values in general, another severe issue is the numerical effort to compute a DNS solution for high Reynolds number turbulent flows. The number of mesh points required (proportional to Re to the 9/4 power) and the corresponding degrees of freedom are extremely large. Thus, in general, such a computation cannot be realized due to computational reqirements and effort. Hence, the question arising is as follows:

Does there exist an appropriate modification of the Navier–Stokes equations such that the modified equations exhibit three conditions: Considering that the system of equations (1a) with \(\mu _t=0\) allows for steady-state solution when the Reynolds number is sufficiently small, for example \(0 < Re \le 10^4\), we follow the idea to decrease (at least locally) the Reynolds number. This goes along with the idea that a proper weighting of the diffusion terms needs to be incorporated to enforce steady-state solutions. From the perspective of (1a), this means that artificially the laminar viscosity \(\mu _l\) needs to be increased. This can be realized by adding an additional function, for example denoted by \(\mu _t\), yielding (8) and (12).

Hence, it is the goal to find a a function (parameter) \(\mu _t\) such that the above mentioned criteria a)–c) are satisfied, and additionally, certain specific flow characteristica are satisfied. In applications these are often for example We summarize these characteristics as given data Y. In general, these data are measurements; and therefore, perturbed by noise. Hence, we assume that instead of Y, there is only noisy data \(Y^{\delta }\) available, which satisfiesIn general, the noise level \(\delta \ge 0\) is unknown.

In this sense, it is the goal of turbulence modeling to prescribe the unknown eddy viscosity \(\mu _t\ge 0\) such that characteristics of interest are represented by a solution to the exterior turbulent flow problem. To give the determination of the unknown function \(\mu _t\) a more general view, we make the following assumption. We assume that \(\mu _{t}(x,t)\ge 0\) and \(\mu _t\) is constant outside of the ball \(B_{d}:=\left\{ x\in {\mathbb {R}}^m: \Vert x\Vert _2\le d\right\} \). The characteristics of interest are often data defined on \(\partial D\), that is, \(Y\in L^2\left( \partial D\right) \). Within this notation we may formulate turbulence modeling as an inverse problem. The forward problem might be stated as follows. Assume that an eddy viscosity (or at least a method to determine \(\mu _t\)) is given, and then find a solution of the exterior turbulent flow problem. Note, in this general formulation, equations implicitly defining \(\mu _t\) are not required any more. One can simply assume the existence and knowledge of \(\mu _t\). Therefore, the finding of a function \(W_t\) would be eliminated in our formulation for the exterior turbulent flow problem.

The inverse problem is to reconstruct the eddy viscosity \(\mu _t\) from given data \(Y\in L^2\left( \partial D\right) \). To formulate the inverse problem mathematically, consider the operatorwhich maps the eddy viscosity to the corresponding data, for example a pressure distribution, such that the exterior turbulent flow problem is satisfied. Here, we assume that the unknown function \(\mu _t\in D(F)\), where D(F) is the domain of the operator F. Naturally, when we assume that given data are velocity profiles, the image space of F needs to be replaced by a data-adapted one.

Though a given turbulence model is an a-priori known methodology to determine \(\mu _t\), calibration of model parameters and validation is performed by comparison with noisy measurement data for a small number of test cases where data is available such that computed data is in good agreement with these measurements. A given turbulence model is a realization of the operator \(F^{-1}\). And the problem is well-posed if F is bijective and \(F^{-1}\) is continuous, otherwise ill-posed.

The interpretation of turbulence modeling as an inverse problem gives rise to several questions, the most severe might be the one of well-posedness. Already, the number of turbulence models in the literature yielding similar reconstructions for data such as the pressure distributions and even skin-friction coefficients suggest non-uniqueness of the problem. Also, a small perturbation in the given pressure distribution may produce a significantly different solution of the exterior turbulent flow problem, and therefore, also a significant difference in the eddy viscosity. This suggests additionally that the condition of continuous dependency on the data is violated. These observations indicate that turbulence modeling by its mathematical nature is ill-posed. Therefore, the finding of a general ansatz to represent characteristics of turbulent flows for a whole variety of problems using the RANS equations in combination with a given methodology determining \(\mu _t\) seems to be challenging, and may even be impossible.

In practice, measurements \(Y^{\delta }\) are available for almost none of the test cases of interest. So, at the moment a practical way for reconstructing \(\mu _t\) requires the knowledge of data independent methods, such as an additional set of equations. Taking into account the possibly ill-posed mathematically nature of the problem, one should be aware of the fact that in general good agreement between noisy measurements and computed data cannot be expected.

Due to the increase in computational power and the ability to realize a scale resolving simulation, we close this section with the following considerations for future work. Assume that a method describing the eddy viscosity \(\mu _{t}\ge 0\) is given, for example an established turbulence model. Let us denote the steady-state solution of the corresponding exterior turbulent flow problem by \(W_{\mathrm{turb}}^{\dagger }\). Furthermore, the associated possible time-dependent solution of the flow problem with \(\mu _t=0\) is given by \(W_{\mathrm{lam}}^{\dagger }\). We introduce the error induced by the turbulence model within the time interval \([T_0,T_1]\) bywhere \(\Vert \; .\;\Vert \) denotes a suitable norm. Such an error measure does not only include surface data but also it includes the entire solution field. We can only expect that \(err_{\mathrm{turb}}\left( T_0, T_1\right) \) can be small if there exists a continuous relationship between the solutions \(W_{\mathrm{lam}}^{\dagger }\) of the laminar and \(W_{\mathrm{turb}}^{\dagger }\) of the Reynolds-averaged Navier–Stokes equations. That is, the change of the solution due to the modification of the set of equations by the eddy viscosity should lead to small differences in the solutions, mathematically speakingThe evaluation of such errors might be helpful to improve the characterization as well as potentials and shortcomings of established models and those developed from scratch.

The argumentation of this paragraph carries over to full Reynolds stress models. Here, instead of reconstructing one parameter \(\mu _t\), six terms defining the Reynolds stress tensor need to be reconstructed.

To confirm that different eddy viscosities can produce almost exactly surface data, we consider a numerical example. For the discretization strategy as well as a solution methodology, we refer to [23, 32‐35]. The software used to generate the results is explained in detail in the literature mentioned.

As proposed in [19], a production limiter is incorporated into the \(k\text {-}\omega \) turbulence models; that is, the production term of the k-equation is replaced by The limiter is constructed such that the value of production does not exceed a multiple of destruction. To demonstrate an effect of these limiters, we consider turbulent flow around the RAE 2822 airfoil with respect to the conditions given in Table 1. The mesh is of size \(320\times 64\). This particular airfoil is quite important since it is frequently considered to evaluate turbulence models. Moreover, there is a substantial amount of detailed data from the experimental investigation of Cook et al. [36]. This data provides the necessary basis for discussing additional issues in this paper.

Figure 4 shows the computed eddy viscosity for Case 9. Note, both results correspond to fully converged solutions. The corresponding convergence histories are given in Fig. 7. The solution without production limiter shows an increase in eddy viscosity in a neighborhood where the shock impinges on the boundary layer. Such an effect is suppressed when using the production limiter, as shown in Fig. 4 (right). The result is reproducible using another implementation [32]. A structured grid code is applied, which uses a different discretization strategy. The results for the computed eddy viscosity are shown in Figs. 5 and 6. Here a different scaling of the plotted values is used.

The same observation is true for Case 10. Given two fully converged solutions (see Fig. 7 (right)), the one with production limiter shows an increase in eddy viscosity in the region where the shock impinges on the surface boundary layer, which vanishes when the production limiter is used.

These numerical results indicate that both test cases have (approximately) no impact on computed \(C_p\) and \(C_f\) distributions. It should be pointed out that essentially the same \(C_p\) variation implies that the boundary-layer displacement thickness is also about the same. Further, since the surface skin-friction variations are essentially the same and the shear stress plays a key role in determining the law-of-the-wall behavior, this suggest that there is a good correlation with the streamwise velocity profiles. A plot of the \(C_p\) and \(C_f\) distributions is given in Fig. 8 for Case 9 and Fig. 9 for Case 10. Obviously, though the eddy viscosity with and without production limiter (68a) is significantly different, its impact on computed \(C_p\) and \(C_f\) distributions is not discernible.

To summarize, this short investigation shows that the introduction of a production limiter (68a) can have significant impact on the computed eddy viscosity. On the other hand, this significant change in the computed eddy viscosity, did not yield significant changes in \(C_p\) and \(C_f\) distributions. Such an observation can be interpreted in two ways. In addition, from the viewpoint of an inverse problem, we need to emphasize that this example reveals that the same data can produce multiple solutions for the eddy viscosity.

Returning to the subject of finding a suitable eddy viscosity \(\mu _t\), we notice that two different functions for \(\mu _t\) generate the same \(C_p\) and \(C_f\) surface distributions. That is, the classical way of turbulence modeling to find a methodology to determine an eddy viscosity such that noisy data is matched cannot verify a unique determination of the eddy viscosity. For example, we could make a numerical experiment where we consider the computed \(C_p\) and \(C_f\) distributions as given and ask for the reconstruction of a corresponding eddy viscosity. Obviously, both computed eddy viscosities, that is the one with and the one without production limiter are solutions of this question. Hence, this example indicates that the inverse problem for these examples does not have a unique answer.

Moreover, we mention that in formulae (68a) and (68b) a factor of 20 was introduced for limitation. However, throughout the literature one finds other factors such as 10 or 5 for limitation of production compared to destruction. The results obtained were similar.

This article has considered boundary-value problems for the RANS equations in combination with two-equation turbulence models. Classical boundary conditions for the unknowns k and \(\omega \) have been discussed, including the assumptions formulated under which these boundary conditions were derived. In so doing, we have assigned a meaning to what one can call a well-defined boundary-value problem. Such a boundary-value problem reveals a characterization of the near-field and far-field solutions of the governing equations. This can be extremely important for providing insight into what to expect when solving the corresponding discrete problem.

For example, it has been shown and discussed that determination of boundary values of \(\omega \) for both the no-slip wall and the far field plays a significant role in determining the behavior at the boundaries for k. Consequently, the behavior of the eddy viscosity \(\mu _t\) in the boundary layer is determined to a large extent by the surface boundary condition for \(\omega \). With respect to this knowledge, it can be assumed that reformulation and manipulation of the equations for the turbulence models will only have limited influence.

In this sense, an improvement of turbulence models might only be possible when significantly better understanding of the complete boundary-value problem is available. Any model including an equation for dissipation rate \(\omega \) is restricted to the shortcomings which are inherent to this equation and its boundary values. In particular, it cannot be excluded that the choice of k and \(\omega \) at the boundaries yields an ill-posed problem.

Furthermore, it has been demonstrated that proposed variable substitutions do not help in general to remove the singular behavior of the equations. The singular behavior is typically only shifted to other terms. Inclusion of a variable substitution gives generally a rise to many more questions than it answers. For example, there is the important question about equivalence of the original boundary-value problem to the one with a substituted variable.

Finally, it has been discussed in which way classical turbulence modeling can be interpreted as an inverse problem, that is, to reconstruct a parameter in a boundary-value problem such that given noisy measurements are approximated. Based on this approach, it has been discussed that the original mathematical nature of turbulence modeling is possibly ill-posed. A numerical example is used to confirm this assertion.