The Baldwin-Lomax Turbulence Model for Two-Dimensional Shock-Wave/Boundary-Layer Interactions

The three-dimensional oblique shock wave-turbulent boundary layer interaction generated by a sharp fin attached to a flat plate is investigated experimentally and theoretically for Mach 3 and Reynolds number 9 x 10 to the 5th using two different models. Both models employ the three-dimensional compressible Navier-Stokes equations in mass-averaged variables; one model utilizes the algebraic turbulent eddy viscosity model of Baldwin and Lomax (1978), while the other model employs the two-equation turbulence model of Jones and Launder (1972) coupled with the wall function model of Viegas and Rubesin (1985). The computed surface pressure, surface streamlines, pitot pressure, and yaw angle profiles are found to be in good agreement with experimental data. The three-dimensional velocity fields computed by both models are in close agreement, although the eddy viscosity profiles differ significantly within the three-dimensional interaction. This result indicates that the overall structure of this three-dimensional sharp fin interaction is insensitive to the turbulence model.

In this paper the development and a partially verification of an anisotropic modification of linear and nonlineare models of turbulence is presented. This modification departs from the observation that there are some regions of fluid domains in which the Reynolds stresses do not depend on the magnitude of the mean flow gradients. The model is based on the hypothesis that actions of fluctuating motion on mean motion are almost exclusively due to large turbulent eddies, and that the latter contain the same kinetic energy of the small eddies of mean flow. The contribution of large turbulent eddies to the Reynolds stresses is then modelled by a linear combination of an isotropic term and an anisotropic term. The latter is obtained by multiplying a quote of the turbulent kinetic energy by a normalised redistribution tensor. The performances of the model are tested calculating the Reynolds stresses in a fully developed turbulent channel flow, and simulating the separated turbulent flow over a backward-facing step, and a fully developed turbulent flow in a squared duct. The results show that the modified models improve the prediction of the normal Reynolds stresses in a bidimensional channel and of the reattachment point in a backward-facing step. On the other hand, these models do not supply any improvement of the secondary flow in a squared duct. INTRODUCTION Algebraic two-equation models of turbulence for Reynolds averaged Navier-Stokes equations are still, today, among the most widely used in the field of engineering design. The two major alternative to such models are the second-order closure models and large-eddy simulations (LES). The weak points of the first approach are high computational cost, due to the five more equations to be solved, and wall-bounded turbulent flows. On the other hand, LES requires much more computational time than second-order closure models, and improved models for the smallscale and the wall region are needed. For these reasons these two approaches have not become yet a useful engineering tool. The aim of this work is to develop an anisotropic modification for linear and nonlinear turbulence model which improves the prediction of normal Reynolds stresses, without increasing the number of differential equations to be solved. More precisely, the objective is to simulate the contribution to the anisotropy of the turbulent stresses observable in areas characterised by low mean velocity gradients. This phenomenon, which can be noted, for example, in experimental data of the turbulent channel flow in figure 5, is not actually described by standard, explicit algebraic models. The starting point of this model consists in the substitution of turbulent kinetic energy by an appropriate vectorial magnitude which provides information about the anistropy of the turbulent stresses. The quantity is identified by the field of velocity of the small eddies of mean motion. The kinetic energy of these small eddies is obtained by assuming that these t Dr, Center of Studies and Activities for Space (CISAS). Dr, Department of Mechanical Engineering. ' Professor, Department of Mechanical Engineering. Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. have the same kinetic energy content of the large turbulent eddies: the two fields of motion are said to be similar. It has been demonstrated that the field of motion of large turbulent eddies, or coherent structures^ can be obtained directly from measurements of instantaneous velocity once the quota of turbulent kinetic energy has been defined. The models obtained provides improvements in the predictions of normal Reynolds stresses present in the turbulent flow of a bidimensional channel; as for the separated turbulent flow in a backward-facing step appreciable improved results of mean flow and of normal Reynolds stresses have been attained compared to standard algebraic models. On the other side, as it will be shown in the following paragraphs, the model of the turbulent kinetic energy redistribution tensor does not supply any improvement in the case of the fully developed turbulent flow in a squared duct. ANISOTROPIC MODIFICATION OF THE REYNOLDS STRESSES Let us consider a field of turbulent motion of a viscous, homogenous and incompressible fluid; the instantaneous velocity v and the instantaneous pressure p can be set out as follows in a mean and fluctuating parts: v = v -f v' p-p + p (1) The evolution of the velocities and mean pressures can be described completely by the well-known Reynolds equations:

A first-order turbulence model based on a turbulent viscosity formulation is derived from a second-order turbulence closure developed for bubbly flows. The interpretation of the second-order turbulence closure in terms of the turbulent viscosity model leads to an analytical analysis of homogeneous bubbly flows. Based on this analysis, a kε turbulence model is then developed and applied to two homogeneous bubbly flows (uniform and with a constant shear). The simulations show that this kε turbulence model that involves a supplementary scale related to the bubble effects succeeds in the representation of the interfacial effects on the turbulence structure in the two homogeneous bubbly flows.

The swept oblique shock-wave/turbulent-boundary-layer interaction generated by a 20-deg sharp fin at Mach 4 and Reynolds number 21,000 is investigated via a series of computations using both conical and three-dimensional Reynolds-averaged Navier-Stokes equations with turbulence incorporated through the algebraic turbulent eddy viscosity model of Baldwin-Lomax. Results are compared with known experimental data, and it is concluded that the computed three-dimensional flowfield is quasi-conical (in agreement with the experimental data), the computed three-dimensional and conical surface pressure and surface flow direction are in good agreement with the experiment, and the three-dimensional and conical flows significantly underpredict the peak experimental skin friction. It is pointed out that most of the features of the conical flowfield model in the experiment are observed in the conical computation which also describes the complete conical streamline pattern not included in the model of the experiment.

Turbulent heat transfer in channel flows is an important area of research due to its simple geometry and diverse industrial applications. Reynolds-Averaged Navier–Stokes (RANS) models are the most-affordable simulation methodology and are often the only viable choice for investigating industrial flows. However, accurate modelling of wall-bounded flows is challenging in RANS, and the assessment of the performance of RANS models for heated turbulent channel flow has not been sufficiently investigated for a wide range of Reynolds and Prandtl numbers. In this study, five RANS models are assessed for their ability to predict heat transfer in channel flows across a wide range of Reynolds and Prandtl numbers (Pr) by comparing the RANS results with respect to the corresponding Direct Numerical Simulation data. The models include three Eddy Viscosity Models (EVMs): standard k−ϵ, low Reynolds number k−ϵLS, and k−ωSST, as well as two Reynolds Stress Models (RSMs): Launder–Reece–Rodi and Speziale–Sarkar–Gatski models. The study analyses the Reynolds number effects on turbulent heat transfer in a channel flow at a Pr of 0.71 for friction Reynolds number values of 180,395,640, and 1020. The results show that all models accurately predict velocity across all Reynolds numbers, but the accuracy of mean temperature prediction drops with increasing Reynolds number for all models, except for the k−ωSST model. The study also analyses the Pr effects on turbulent heat transfer in a channel flow with Pr values between 0.025 and 10.0. An error analysis is performed on the results obtained from different turbulence models, and it is shown that the k−ωSST model has the smallest error for the predictions of the mean temperature and Nusselt number for high-Prandtl-number flows, while the low Reynolds number k−ϵLS model shows the smallest errors for low-Prandtl-number flows at different Reynolds numbers. An analytical solution is utilised to identify Pr effects on forced convection in a channel flow into three different regimes: analytical region, transitional region, and turbulent diffusion-dominated region. These regimes are helpful to discuss the validity of the models in relation to the Pr. The findings of this paper provide insights into the performance of different RANS models for heat transfer predictions in a channel flow.

Abstract. In this paper we investigate the finite-time and asymptotic behaviour of algebraic turbulent mixing-layer models by numerical simulation. We compare the performances given by three different settings of the eddy viscosity. We consider Richardson number-based vertical eddy viscosity models. Two of these are classical algebraic turbulence models usually used in numerical simulations of global oceanic circulation, i.e. the Pacanowski–Philander and the Gent models, while the other one is a more recent model (Bennis et al., 2010) proposed to prevent numerical instabilities generated by physically unstable configurations. The numerical schemes are based on the standard finite element method. We perform some numerical tests for relatively large deviations of realistic initial conditions provided by the Tropical Atmosphere Ocean (TAO) array. These initial conditions correspond to states close to mixing-layer profiles, measured on the Equatorial Pacific region called the West-Pacific Warm Pool. We conclude that mixing-layer profiles could be considered as kinds of "absorbing configurations" in finite time that asymptotically evolve to steady states under the application of negative surface energy fluxes.

The accurate modelling of heat transfer to turbulent flow and the prediction of the temperature distribution in the flow remain one of the problem areas of numerical simulations. Traditional turbulence closure models, like the k–ε model, effectively only increase the viscosity of the fluid and introduce wall functions close to boundaries to obtain the correct velocity distribution. These turbulence models do not model the small‐scale mixing that occurs in turbulent flow. When solving the energy equation these small‐scale mixings dominate the heat transfer rate at the boundaries as well as the temperature distribution in the flow. This paper outlines a revised method, based on the k–ε turbulence model, that can be used to predict heat transfer in turbulent flow. A single turbulent conductivity term is introduced that can be used over the complete flow field including the boundaries. A detailed description of the mathematical model and boundary conditions used for the turbulence model are included in the paper. The effective turbulent conductivity method was evaluated in several finite difference simulations of water flowing through a smooth pipe while being heated. Simulation and verification were performed over a range of Reynolds numbers. Verification of the model is accomplished by comparing the numerically predicted centre temperature of the fluid as well as the heat flux to the fluid to measured temperatures in a similar pipe. From these results it is concluded that the revised turbulent conductivity model holds great potential to obtain accurate simulated heat transfer rates for general applications.

A computational study is presented for the dynamic stall of an airfoil that is pitched at a constant rate from zero incidence to a high angle of attack. The unsteady flow is simulated employing the mass-averaged NavierStokes equations and an algebraic turbulent eddy viscosity model. The approach is first validated by comparison of computed and experimental results for a pitching airfoil at low freestream Mach numbers. The computed dynamic stall events, as well as the computed effects of pitch rate and axis location, are found in qualitative agreement with experimental observations. The effect of compressibility on dynamic stall is investigated. As the freestream Mach number increases, the appearance of a supersonic region provides—through the shock/boundarylayer interaction—an additional mechanism in the dynamic stall process. The main effects of compressibility are found to be 1) a change from trailing-edge stall to leading-edge stall and 2) a reduction in the stall delay and in the attained maximum lift.

: An investigation of five computer programs which calculate skin friction and boundary-layer integral properties in two-dimensional, turbulent, transonic flow was conducted. The five programs were of varying complexity and consisted of four boundary-layer-type methods with turbulence modeling varying from algebraic eddy viscosity models to a two-equation of turbulence model, and one mean flow compressible Navier-Stokes technique with algebraic turbulence modeling. The quality of the computed results was judged by comparison with available experimental data. For the data considered (which were nonseparated, two-dimensional, transonic, adiabatic wall data), the most accurate calculations of skin friction were provided by an integral boundary-layer method and a finite difference boundary-layer method, both with algebraic turbulence modeling.

Numerical investigation of heat transfer performance and flows characteristics in turbine blade internal cooling using Pin-Fin arrays coupled with discontinuous ribbed endwall

Using conjugate heat transfer calculations, the heat transfer in straight-through labyrinth seals with and without rub damages (bending and mushrooming damages) were numerically investigated. Firstly, the numerical methods were carefully validated on the basis of obtained experimental data. At two different sealing clearances and a range of Reynolds numbers, Nu distributions on the seal rotor and stator surfaces for the original design cases were numerically computed and compared to the experimental data. The temperature fields in the fluid and inside the solid domains were obtained to account for the heat transfers between fluid and adjacent solids. Then, a range of bending angles, wear-off ratios and mushrooming radiuses were selected to investigate the influence of rub damages on heat transfer characteristic in the labyrinth seals, and the numerical results were also compared to that of original design cases. The results show that the calculated Nu distributions are in good agreement with the experimental data at a range of Re numbers and different sealing clearances. The turbulence model has pronounced effect on the heat transfer computations for the labyrinth seal. Among the selected eddy viscosity turbulence models, the low-Re k-ω and SST turbulence models show superior accuracy to the standard k-ε and RNG k-ε turbulence models, which over-predict Nu by about 70%. Bending damage reduces Nu on the labyrinth fin whereas enhances heat transfer on the opposite smooth stator. The effect of bending angle on Nu distribution on the seal stator surface is larger than on the rotor surface. The mushrooming damage has pronounced effect on Nu distributions on both rotor and stator surfaces for the labyrinth seal. It shows that Nu distributions on the rotor and stator surfaces decreases with the increase of mushrooming radius, but increases with the increase of wear-off ratio and Re.

Reynolds-averaged turbulence models have become one of the most important and popular techniques for practical engineering applications in aeronautics and astronautics. However, the poor performance in the prediction of flow separations restricts its application ranges due to the traditional linearity and equilibrium hypotheses that constitute the equation of Reynolds stress in turbulence modeling. In this study, an artificial neural network-based quadratic constitutive (ANN-QCR) Reynolds stress model is proposed for simulating turbulent flows with separations by using the field inversion and machine learning technique (FIML) with high-fidelity experimental data. In particular, the Reynolds stress is decomposed into linear and non-linear parts, respectively. The former is evaluated by the Spalart–Allmaras model with a correction factor imposed on the production term to account for the non-equilibrium effect, while the latter is a quadratic constitutive term with a self-calibrated factor. These correction factors are predicted by an artificial neural network (ANN) depending on the local flow features. The unified framework of FIML updates the weights of ANN-QCR directly by the gradient-based discrete adjoint method, ensuring consistency between the field inversion and the neural network training. The data-augmented ANN-QCR turbulence model is well validated through several separated turbulent flows induced by adverse pressure gradients, shock wave boundary interfaces, higher angles of attack, and higher Reynolds numbers (Re). With the optimization target at lift coefficients, the established model also improves the predictive performance in other flow quantities, such as drag coefficients and pressure distributions. In addition, this model captures the development of separation bubbles better with the increase in the angle of attack. Benefiting from the compatibility and convergence of forward simulation, the generalization capability of the present ANN-QCR model is successfully validated in various numerical simulations of separated turbulent flow problems across a wide range of attack angles and Reynolds numbers with good accuracy.

The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-

A combined theoretical and experimental study is presented for the interaction between crossing shock waves generated by (10°, 10°) sharp fins and a flat plate turbulent boundary layer at Mach 8.3. The theoretical model is the full 3-D mean compressible Reynolds-averaged Navier-Stokes RANS) equations incorporating the algebraic turbulent eddy viscosity model of Baldwin and Lomax. A grid refinement study indicated that adequate resolution of the flowfield has been achieved. Computed results agree well with experiment for surface pressure and surface flow patterns and for pitot pressure and yaw angle profiles in the flowfield. The computations, however, significantly overpredict surface heat transfer. Analysis of the computed flowfield results indicates the formation of complex streamline and wave structures within the interaction region.

Aircraft with slender bodies and swept lifting surfaces, flying at high angles of attack, often have regions of three-dimensi onal separated flow characterized by strong off-body vortices. One approach to numerically simulate this class of high Reynolds number flows is to use the Reynolds-averaged Navier-Stokes equations (RANS), together with a turbulence model. Many turbulence models have been utilized for the RANS equations, however most are developed and calibrated for configurations with attached boundary layers or flows with shockinduced separation. In order to apply these models to separated flows with strong off-body vortices, modifications are usually necessary. This investigation examines the development and application of these modifications for both the algebraic Baldwin-Lomax turbulence model, and the one-equation Spalart-Allmaxas model. The implementation of these modified models differs among several production computational fluid dynamics (CFD) codes, as do their computed results. Model behavior in predicting crossflow separation for an ogive cylinder, and leadingedge separation for a delta wing are examined. The flowfields for the ogive cylinder and delta wing geometries are simulated at 20° and 15° angle of attack respectively. Both configurations are computed at subsonic flow speeds, however applications to transonic and supersonic wingbody configurations are discussed. Comparisons are made with experimental surface pressure measurements, and surface oil-flow visualizations where available.