“It’s easy to explain how a rocket works, but explaining how a wing works takes a rocket scientist…” – Philippe Spalart
Most of nowadays CFD simulations are conducted with the Reynolds Averaging approach. Reynolds-Averaged Navier-Stokes (RANS) simulation is based on the Reynolds decomposition according to which a flow variable is decomposed into mean and fluctuating quantities. When the decomposition is applied to Navier-Stokes equation an extra term known as the Reynolds Stress Tensor arises and a modelling methodology is needed to close the equations. The “closure problem” is apparent as higher and higher moments of the set of equations may be taken, more unknown terms arise and the number of equations never suffices. This is of course an obvious consequence to the fact that taking these higher moments is simply a mathematical endeavor and has no physical contribution what so ever.
Levels of modeling are related to the number of differential equations added to Reynolds Averaged Navier-Stokes equations in order to “close” them.
0-equation (algebraic) models are the simplest form of turbulence models, a turbulence length scale is specified in advance through experimenting. 0-equations models are very limited in applications as they fail to take into account history effects, assuming turbulence is dissipated where it’s generated, a direct consequence of their algebraic nature.
1-equation and 2-equations models, incorporate a differential transport equation for the turbulent velocity scale (or the related the turbulent kinetic energy) and in the case of 2-equation models another transport equation for the length scale (or time scale), subsequently invoking the “Boussinesq Hypothesis” relating an eddy-viscosity analog to its kinetic gasses theory derived counterpart (albeit flow dependent and not a flow property) and relating it to the Reynolds stress through the mean strain.
In this sense 2-equation models can be viewed as “closed” because unlike 0-equation and 1-equation models (with exception maybe of 1-equations transport for the eddy viscosity itself) these models possess sufficient equations for constructing the eddy viscosity with no direct use for experimental results.
RANS differential equation closure models do however contain many assumptions along the way for achieving the final form of the transport equations and as such are calibrated to work well only according to well-known features of the applications they are designed to solve. Nonetheless although their inherent limitations. Nonetheless the modeling methodology strength has proven itself for wall bounded attached flows at high Reynolds number (thin boundary layers) due to calibration according to the “law-of-the-wall”.
As RANS is already a working horse but has shown poor performance due its inherent limitation applied to flows of which strong instabilities and large unsteadiness occurs and it does not seem that a breakthrough in achieving a universal modeling methodology is expected soon (or at all…), researchers have shifted much of the attention and effort to hybrid formulations incorporating RANS and LES in certain ways.
The Spalart-Allmaras model serves also as the basis for the formulation of perhaps the most popular hybrid simulation model, Detached Eddy Simulation (DES)
Eddy Viscosity Transport Equation Turbulence Models
Eddy viscosity transport equation turbulence models are a special kind of 1-equation models in as much that they posses completeness (one differential equation for the eddy viscosity to be related directly to the Reynolds Stresses). It is actually very straightforward to derive an eddy viscosity transport equation from a 2-equation closure. As an example, I shall present such a simple procedure to be followed (I shall call it “my 1-eq eddy viscosity transport closure stunt” 😉 ):
The simple derivation above incorporates some interesting features as the “Bradshaw hypothesis” and the Von-Karman length scale, alterations that are important for shear stress transport and unsteadiness (not to say that they are actually exploited by this specific model).
Although it was very simple to derive, the calibration of the constants shall be not less important to achieve a physically meaningful turbulence model or specifically tuned for the range of validity in flows of interest. I shall return to that in the following paragraphs.
The Original Spalart-Allmaras Turbulence Model
The Spalart-Allmaras Turbulence Model has been developed mainly for aerodynamic ﬂows. The formulation blends from a viscous sublayer formulation to a logarithmic formulation based on y+. In as such, no addition of highly non-linear damping functions for laminar/viscous sublayer modeling is in use.
The methodology for obtaining the 1-equation eddy viscosity transport model was somewhat different from the usual derivation of 2-equation models, regularly achieved by pure mathematical operations to be subsequently simplified by physical reasoning.
As for every transport equation, such is for the Reynolds stresses and after Relating the eddy viscosity to the mean strain rate:
The various terms in an equation for the transport of the eddy viscosity can be identified as convection, diffusion, production and destruction.
In the spalart-allmaras methodology, surgical physics based assumptions were made concerning each of the various terms such as diffusion, production and destruction to the final aim of achieving a complete transport equation for the eddy viscosity:
Spalart-allmaras representation of the diffusion terms kicks off with the classical diffusion operator as ∇·([νt/σ]∇νt) where σ is a turbulent Prandtl number. Then, as to achieve an aerodynamic flow oriented diffusion behavior it is pointed out that there is no reason for the integral of the eddy viscosity to be conserved due to cross terms between ∇k and ∇ε for example (as seen from the above derivation emanating from a 2-equation model), hence a non-conservative term is added to the classical diffusion description:
The representation of the production term is analogue to the production of turbulent energy, assuming that it shall rise with an increase in total viscosity and with the increase of the mean vorticity. The subtlety is in the consideration of the relation between the production and which form of mean vorticity is to be chosen in the quest for most favorable effect. In the case of Spalart-Allmaras the favorable effect is to serve aerodynamic ﬂows in which turbulence is found only where vorticity is. Therefore the magnitude of vorticity is chosen as the representation of the mean vorticity (it should be noted that future advancements of the original model to account for various flow features may consider other forms for its representation) and the production takes the form:
As for destruction, it is assumed for the eddy viscosity that “the ability of a turbulent ﬂow to transport momentum and the ability must be directly related to the general ”level of activity“, therefore to the turbulent energy to construct the destruction term” (taken from Nee and Kovasznay – 1969). It’s also claimed in the surgical process of deriving the destruction term that there is a “blocking effect” from a wall that is felt at a distance by the pressure and acts as a destruction entity for the Reynolds shear stress, therefore the use of a wall proximity parameter in the representation is mandatory under these assumptions (as a side note, the wall proximity parameter shall serve well in constructing future hybrid DES and especially its advancements as described in my former post: Detached Eddy Simulation – an attractive methodology to RANS in the aid of LES ).
The above assumption, and under proper calibration of the constants seem to reproduce an accurate log layer. On the other hand, the skin friction it produces for a flat plate boundary layer is underestimated, a consequence of the rate of decay of the destruction term in the outer portion of the boundary layer. Therefore, the Spalart-Allmaras destruction term contains a function (which is equal to 1 in the log-layer) to control this rate of decay and takes the final form of the destruction representation:
where d is the wall proximity parameter, cw1 the constant to be calibrated and fw is the control function.
Subsequently to performing the surgical identification of the different terms in the transport equation for the eddy viscosity (and by relation the Reynolds Stress), it should be remembered that we are still left out with some added constants to be calibrated and a control function to be matched. In turbulence modeling calibration of the model is at least as important as the derivation of the model itself. Calibration is achieved with the help of experimental and numerical results of the type of ﬂow that should be modeled. The calibration process is also the first step in which the range of validity of the model would be revealed to close inspection and not just postulated from physical reasoning.
Saying all that, the process of calibrating the model is much more complicated than the description of the main steps given below.
To achieve the non-dimensional control function Spalart-Allmaras route followed the 0-equation mixing length wall interaction methodology such that the function behavior shall present satisfactory prediction of the problematic outer-layer of the boundary layer. Limiting procedure to prevent large value of the control function which could be problematic for the numerical simulations are also an inseparable part of the derivation.
After the surgical considerations for the control function supported by experimental results to also achieve calibration of the constants we are left with the following:and for the control function (the phase from g to the control function represents the limiter effect):
The second (length scale), third (strain rate tensor) and fourth (friction velocity) terms in the above are based on assumptions of classical log-layer definitions which allows to assume equilibrium between production, diffusion and the destruction as long as the first term is defined as such.
The Final Form of Spalart-Allmaras Turbulence Model
Following the surgical derivation of the transport equation, the calibration of its constants and the identification of the features and form for the control function one of the most popular turbulence models for aerodynamic calculations final form may be presented:
Modification of Spalart-Allmaras Closure to Acount for Rotation and Curvature
A drawback evident in almost all eddy-viscosity models is the inability to inherently account for rotation and curvature. This drawback is resulted from relating the Reynolds stress to the mean flow strain and in fact is the major difference between such a modeling approach and a full Reynolds-stress model (RSM). The RSM approach accounts for the important effect of the transport of the principal turbulent shear-stress. On the other hand, RSM simulations are not computationally cost-effective, in as much that one does get an improved physical fidelity that is worth the time and computational resources consumed, not only that, they often do not converge.
In the aim of accounting for rotation and curvature effect Spalart and Shur offered a modification for the model based on empirical grounds.
The route for altering the transport equation kicks off with the identification of the effect of curvature and rotation in two types of extreme flows:
- thin shear flows with weak rotation (compared with the shear rate) or weak curvature (compared with the inverse of the shear-layer thickness), highly impacting the level of the turbulent shear stress.
- homogeneous rotating shear flow and free vortex cores of which strong rotation reduces the turbulent shear stress sharply.
Spalart offers an alteration of the original eddy viscosity transport equation based on reasoning concerning the first type presented above (thin shear flows) and another empirical alteration added to the former to account for the second type of extreme (i.e. homogeneous rotating shear flow and free vortex cores).
The physical reasoning returns yet again to the chosen way to regard the eddy viscosity relation to the strain rate and vorticity in a way as to alleviate non-addressing the discrepancy between the principal axes of the Reynolds stress tensor and rate of strain tensor.
Subsequently to exploring a satisfactory relationship between strain-rate and vorticity to be accounted for by a scalar quantity to handle curvature and rotation for thin shear flows with weak rotation and also (albeit less reliably but could be improved by empirical additives) for homogeneous rotating shear flow and free vortex cores, the production term of the eddy viscosity transport equation is multiplied by a “rotation function” (constants Cr1, Cr2 and Cr3 are calibrated as 1, 12 and 1 respectively) :