Baldwin-Lomax Model Research Articles

An improved Baldwin–Lomax algebraic wall model for high-speed canonical turbulent boundary layers using established scalings

In this work, we employ well-established relations for compressible turbulent mean flows, including the velocity transformation and algebraic temperature–velocity (TV) relation, to systematically improve the algebraic Baldwin–Lomax (BL) wall model for high-speed zero-pressure-gradient air boundary layers. Any new functions or coefficients fitted by ourselves are avoided. Twelve published direct numerical simulation (DNS) datasets are employed for a priori inspiration and a posteriori examination, with Mach numbers up to 14 under adiabatic, cold and heated walls. The baseline BL model is the widely used one with semilocal scalings. Three targeted modifications are made. First, we employ a total-stress-based transformation (Griffin et al., Proc. Natl Acad. Sci. USA, vol. 118, issue 34, 2021, e2111144118) to the inner-layer eddy viscosity for improved scaling up to the logarithmic region. Second, we utilize the van Driest transformation in the outer layer based on the compressible defect velocity scaling. Third, considering the difficulty in modelling the rapidly varying and singular turbulent Prandtl number near the temperature peak in cold-wall cases, we design a two-layer strategy and use the TV relation to formulate the inner-layer temperature. Numerical results prove that the modifications take effect as designed. The prediction accuracy for mean streamwise velocity is notably improved for diabatic cases, especially in the logarithmic region. Moreover, a significant improvement in mean temperature is realized for both adiabatic and diabatic cases. The mean relative errors of temperature to DNS for all cases are down to 0.4 % in the logarithmic wall-normal coordinate and 3.4 % in the outer coordinate, around one-third of those in the baseline model.

On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations

In this paper we consider the steady Baldwin-Lomax model, which is a rotational model proposed to describe turbulent flows at statistical equilibrium. The Baldwin-Lomax model is specifically designed to address the problem of a turbulent motion taking place in a bounded domain, with Dirichlet boundary conditions at solid boundaries. The main features of this model are the degeneracy of the operator at the boundary and a formulation in velocity/vorticity variables. The principal part of the operator is non-linear and it is degenerate, due to the presence (as a coefficient) of a power of the distance from the boundary: This fact makes the existence theory naturally set in the framework of appropriate weighted-Sobolev spaces.

Extension of a simplified Baldwin–Lomax model to nonequilibrium turbulence: Model, analysis and algorithms

AbstractComplex turbulence not at statistical equilibrium is impossible to simulate using eddy viscosity models due to a backscatter. This research presents the way to correct the Baldwin–Lomax model for nonequilibrium effects and gives an analysis of the energy evolution in the corrected model. Furthermore, a finite element approximation of the corrected eddy model with first‐order and second‐order time discretization is also presented. A numerical test is given to support the theory.

Analysis of the turbulent boundary layer on a flat plate at M=6÷8.8 with the use of NERAT-2D code and algebraic turbulence models

Experimental data on heat transfer of supersonic air flows to flat plate, which were analysed by Dilley A.D [NASA/CR-2001-210837, 2001], are compared for Mach numbers M = 6 ÷ 8.8 with the calculated data obtained using the Navier – Stokes flow model in conjunction with the analytical models of turbulent mixing of the Prandtl mixing length and the Baldwin-Lomax model. In the paper, the author's computer code NERAT-2D was used for numerical study.

Numerical calculations of 2D transonic flow in GAMM channel and over the profile

The aim of this work is to make a 2D numerical model of the solution of the transonic inviscid and viscous compressible flow around the profile. In a case of viscous flow several turbulent models are used. For the verification of the calculation Baldwin-Lomax model is compared with Wilcox k-omega model and SST turbulent model. Calculations are done in GAMM channel computational domain with 10% DCA profile and in the turbine cascade computational domain with 8% DCA profile. Numerical methods are based on a finite volume solution. Comparisons are done with the experimental data for the 8% DCA profile.

Numerical solution of inviscid and viscous laminar and turbulent flow around the airfoil

This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic Baldwin-Lomax model is used and compared with Wilcox k-omega model. Calculations are done for NACA 0012 and RAE 2822 airfoil profile for the different angles of upstream flow. Numerical results are compared and discussed with experimental data.

Property evaluation on turbulence models calculation in hypersonic heat transfer simulation

Factors that influence the prediction accuracy of hypersonic heat transfer simulation under turbulence were studied with the double ellipsoid model by the means of computational fluid dynamic. Four popular turbulent models,which include the Baldwin-Lomax( BL) model,the Spalart-Allmaras( SA) model,the Wilcox k-ω model and the Menter shear-stress transport( SST) model,were employed in the study of heat transfer simulation,their performances were assessed,the dependence of heat transfer on grid Reynolds number was assessed. Besides,the connection between sweep angle and the heat in stagnation point was established by simulation with seven different sweep angles in blunt fin. The results indicate that the Two equation model,especially the SST model,is more suitable for the heat transfer simulation in reattaching flow than the One equation model and the Zero equation model. The results also show that the maximum heat of blunt fin varies non-monotonic with the sweep angle and reaches a peak value when the sweep angle is about 22°.

Numerical Simulation of Airfoil Flow at High Angle of Attack

A NACA 4412 airfoil tested in wind tunnel at flow conditions close to maximum lift is used for testing the accuracy of various turbulence models. What makes this test case unique is the appearance of a stable separation vortex on the upper surface of the airfoil, near the trailing edge. The linear k-ω turbulence model, a non-linear Explicit Algebraic Stress Model and a modified version of the algebraic Baldwin-Lomax model are tested using the same grid and input file. It is found that the tested turbulence models capture the physics of unsteady separated flow. However, the size and shape of the predicted separation vortex are different for each tested turbulence model. Also, good agreement between computational and experimental surface pressures is observed. The results support the view that such a simple configuration is appropriate to be used as benchmark for validating turbulence and LES models.

Numerical solution of inviscid and viscous flow around the profile

This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic Baldwin-Lomax model is used and compared with Wilcox’s k-ω model. Calculations are done in GAMM channel computational domain with 10% DCA profile and in turbine cascade computational domain with 8% DCA profile. Numerical methods are based on a finite volume solution and compared with experimental measurements for 8% DCA profile.

High-order discontinuous Galerkin solver on hybrid anisotropic meshes for laminar and turbulent simulations

Efficient and robust solution strategies are developed for discontinuous Galerkin (DG) discretization of the Navier-Stokes (NS) and Reynolds-averaged NS (RANS) equations on structured/unstructured hybrid meshes. A novel line-implicit scheme is devised and implemented to reduce the memory gain and improve the computational efficiency for highly anisotropic meshes. A simple and effective technique to use the modified Baldwin-Lomax (BL) model on the unstructured meshes for the DG methods is proposed. The compact Hermite weighted essentially non-oscillatory (HWENO) limiters are also investigated for the hybrid meshes to treat solution discontinuities. A variety of compressible viscous flows are performed to examine the capability of the present high-order DG solver. Numerical results indicate that the designed line-implicit algorithms exhibit weak dependence on the cell aspect-ratio as well as the discretization order. The accuracy and robustness of the proposed approaches are demonstrated by capturing complex flow structures and giving reliable predictions of benchmark turbulent problems.

Improvement of Baldwin–Lomax turbulence model for supersonic complex flows

Entropy represents the dissipation rate of energy. Through direct numerical simulation (DNS) of supersonic compression ramp flow, we find the value of entropy is monotonously decreasing along the wall-normal direction no matter in the attached or the separated region. Based on this feature, a new version of Baldwin–Lomax turbulence model (BL-entropy) is proposed in this paper. The supersonic compression ramp and cavity-ramp flows in which the original Baldwin–Lomax model fails to get convergent solutions are chosen to evaluate the performance of this model. Results from one-equation Spalart–Allmaras model (SA) and two-equation Wilcox k–ω model are also included to compare with available experimental and DNS data. It is shown that BL-entropy could conquer the essential deficiency of the original version by providing a more physically meaningful length scale in the complex flows. Moreover, this method is simple, computationally efficient and general, making it applicable to other models related with the supersonic boundary layer.

Numerical experiments for turbulent flows

The aim of the work is to explore the possibilities of modelling transonic flows in the internal and external aerodynamics. Several configurations were analyzed and calculations were performed using both inviscid and viscous models of flow. Viscous turbulent flows have been simulated using either zero equation algebraic Baldwin-Lomax model and two equation k—ω model in its basic version and improved TNT variant. The numerical solution was obtained using Lax-Wendroff scheme in the MacCormack form on structured non-ortogonal grids. Artificial dissipation was added to improve the numerical stability. Achieved results are compared with experimental data.

Predication of 3-D Viscous Flowfield of a Centrifugal Impeller

A three-dimensional viscous code has been developed to solve Reynolds-averaged Navier-Stokes equations. The governing equations in finite volume form are solved by two-step Runge-Kutta scheme with implicit residual smoothing. The eddy viscous is obtained using the Baldwin-Lomax model. A prediction of the 3-D turbulent flow and the performance in the “all-over controlled vortex distribution” centrifugal impeller with a vaneless diffuser has been made for the compressor at design and off-design condition. The predicted effi-ciency is a little higher than the experiment data. These results suggest that the present calculation code is able to determine the flow development in the impeller and also the turbulence model in the centrifugal im-peller should be improved.

Numerical Study of Heat Transfer Around the Small Scale Airfoil Using Various Turbulence Models

A numerical investigation of low-Reynolds number flows with thermal effect around the MAV airfoils using various turbulence models, including an algebraic Baldwin-Lomax model, Spalart-Allmaras one equation, and two equation (k-ω and SST-k-ω) turbulence models, is presented. First, the thermal effect on the aerodynamic efficiency is studied for flow around a rectangular MAV wing, based on the NACA0012 airfoil section at low-aspect ratio (AR = 2) and an angle of attack equal to 0°. Second, details of the thermal effect are limited to the two-dimensional NACA0012 airfoil with chord length of 3.81 cm. This study shows that the improvement of aerodynamic efficiency (increase lift and reduce drag) is achieved by the generation of a temperature difference between extrados and intrados of the airfoil (by cooling the upper surface and heating the lower surface). The numerical results obtained with various turbulence models are in good agreement with experiment data, except the k-ω turbulence model. These results are performed with the CFD-FASTRAN code, using the fully implicit scheme for time integration and the upwind Roe flux difference splitting scheme for space discretization augmented by a high order Osher-Chakravarthy limiter.

Turbulent flow around single concentric long capsule in a pipe

A numerical solution was developed for the equations governing the turbulent flow around single concentric long capsule in a pipe. First, a turbulence model was established for the concentric annulus between the capsule and the pipe to simulate the flow as axi-symmetric, two dimensional, steady flow without edge effect. Second, the same case was considered taking into account the edge effect. Finally, turbulence modelling was established to simulate the case as a three dimensional steady flow, with a view of investigating the validity of axi-symmetric flow assumption. Three different turbulence models were used: an algebraic model (Baldwin–Lomax model) and two types of two-equation models ( k– ε and k– ω). Obtained results of pressure gradient along the capsule were compared with available experimental data to verify the used models. In addition, experimental data of the velocity profiles of other investigators were also used in this concern. The results predicted by the three different turbulence models were shown to agree well with the experimental data, though precision differed from one to another.

An inverse design method for viscous flow in turbomachinery blading using a wall virtual movement

An inverse shape design method for turbomachinery blades based on a time-accurate solution of the viscous flow equations is presented. The design scheme is formulated such that either the blade pressure distributions on pressure and suction surfaces, or the blade pressure loading and its thickness distribution can be prescribed as design variables. The blade profile is modified using a virtual velocity distribution that would make the momentum flux on the blade surfaces equal to the design momentum flux. The flow is simulated by solving the Reynolds-averaged Navier–Stokes equations that are discretized using a cell vertex finite-volume method, where an arbitrary Lagrangian–Eulerian formulation is used to account for mesh movement. An algebraic Baldwin–Lomax model is used for turbulence closure. The inverse method is first validated for a transonic compressor cascade; it is then used to redesign a subsonic turbine and a transonic compressor. The results show that the design method is rather robust, flexible and useful in reshaping the blade geometry to achieve the prescribed design variables. They also indicate that by carefully tailoring the design target, significant improvement can be achieved in the blade aerodynamic performance.

A combined experimental and numerical study of flow structures over three-dimensional shock control bumps

A detailed study of transonic flow over three-dimensional bumps has been conducted using experimental measurements and computational simulation. The aim of the investigation is to determine the flow characteristics over these potentially effective flow control devices for wave drag reduction for transonic aircraft wings. Careful qualitative and quantitative matching of the simulation and experimental conditions has considerably improved the agreement between them. In both the experiments and the computation, the shock position in the working section was found to be sensitive not only to the back pressure and the sidewall effects, but also to the incoming boundary layer characteristics. Two turbulence models, a modified Baldwin–Lomax model with enhanced performance in separated flow (curvature model) and a two-equation model ( k – ω ) have been implemented in the numerical simulation. Overall, both turbulence models gave reasonable results for the uncontrolled and controlled cases regarding the inviscid flowfield and the shock structure. In particular, a pair of streamwise vortices embedded in the boundary layer was captured by the two models. The traces in the surface oil flow pictures from the experiment also suggested the existence of the streamwise vortices. The combined surface and flowfield data provide some further insight into the flow physics on the shock control ramp bump, which is discussed in the paper. In addition, the study also demonstrates the enhanced capability of the algebraic model in capturing separated flow features with lower computational cost as compared to the k – ω model.

Dynamic hybridization of MILES and RANS for predicting airfoil stall

Hybridization comprised of an algebraic turbulence model based on the Reynolds average Navier–Stokes (RANS) equations with a monotonically integrated large eddy simulation (MILES) is proposed to simulate transient fluid motion during separation and vortex shedding at high Reynolds numbers. The proposed hybridization utilizes the Baldwin–Lomax model with the Degani–Schiff modification as the RANS model in the near-wall region and a MILES far from the wall. Although the hybridization is assumed to be a MILES with wall modeling, the transition line between the RANS and the MILES modes is determined by the turbulent intensity that is dominated by the large eddies in the grid-scale. This hybrid model is applied to the flows past three different types of airfoils (NACA63 3-018, NACA63 1-012 and NACA64A-006) near stall, at a chord Reynolds number of Re = 5.8 × 10 6. These airfoils are classified as trailing-edge-stall, leading-edge-stall and thin-airfoil-stall airfoils, respectively. The computed results are compared with wind tunnel experiments. The hybrid model successfully demonstrates accurate stall angle and surface pressure distribution predictions near the stall for each type of airfoil. The airfoil simulation results confirmed that the hybrid model provides a better prediction than the RANS model for unsteady turbulent flows with separation and vortex shedding simulations.

Numerical simulation of thin airfoil stall by using a modified DES approach

AbstractThe detached‐Eddy simulation (DES) method was applied to calculate pre‐ and post‐stall aerodynamic characteristics of airfoil stall. A discrepancy between numerical and experimental data was observed near the stall regime for the airfoil NACA64A‐ 006 which is a thin airfoil stall type. The reason of this discrepancy and one possible way for improvement of the numerical model are discussed here. It is shown that the use of the Baldwin–Lomax model in the RANS region improves the DES results in this case. If the relevant factors (grid density, time step, turbulence model, etc.) are appropriately taken into account, the DES approach could reliably predict stall aerodynamical characteristics. Copyright © 2006 John Wiley & Sons, Ltd.

Numerical analysis of unsteady behavior of cloud cavitation around a NACA0015 foil

Three-dimensional unsteady cavitating flow around a NACA0015 hydrofoil fixed between the sidewalls was simulated and the mechanism of U-shaped cloud cavity formation was clarified. A local homogeneous model was used for the modeling of the vapor–liquid two-phase medium. The compressible two-phase Navier–Stokes equations as the governing equations were solved. To describe the phase change between water and vapor, the mass transfer model based on the theory of evaporation/condensation on a plane interface was introduced. The cell-centered finite volume method was employed to discretize the governing equations. Assuming turbulent flow, the turbulent eddy viscosity coefficient was computed by using the Baldwin–Lomax model with the Degani–Schiff modification. As a result, even in the case of cavitating flow without sidewalls, the shed cloud cavities has slightly 3D structure, which was not so much large as extending across the whole spanwise direction. On the other hand, in the case of cavitating flow with sidewalls, the end of sheet cavities bows in the spanwise direction because of the development of boundary layer near both sidewalls. After that, due to the occurring of the reentrant jet towards the mid-span region, the sheet cavities breaks off from mid-span region near the leading edge of the hydrofoil, and became the vortical cloud cavities, which have the large-scale U-shaped structure.