Algebraic Turbulence Model Research Articles

A numerical study of shock wave/boundary layer interaction in a supersonic compressor cascade

A numerical analysis of shock wave/boundary layer interaction in transonic/supersonic axial flow compressor cascade has been performed by using a characteristic upwind Navier-Stokes method with various turbulence models. Two equation turbulence models were applied to transonic/supersonic flows over a NACA 0012 airfoil. The results are superion to those from an algebraic turbulence model. High order TVD schemes predicted shock wave/boundary layer interactions reasonably well. However, the prediction of SWBLI depends more on turbulence models than high order schemes. In a supersonic axial flow cascade at M=1.59 and exit/inlet static pressure ratio of 2.21, k-μ and Shear Stress Transport (SST) models were numerically stables. However, the k-μ model predicted thicker shock waves in the flow passage. Losses due to shock/shock and shock/boundary layer interactions in transonic/supersonic compressor flowfields can be higher losses than viscous losses due to flow separation and viscous dissipation.

Towards the development of a Reynolds-averaged algebraic turbulent scalar-flux model

Abstract In order to derive a possible direction for developing Reynolds-averaged algebraic turbulent scalar-flux models, a priori explorations are attempted by processing the LES data presently performed for channel flows under several flow-boundary conditions including shear-free boundaries. The present calibration has elucidated that the turbulent scalar-flux vectors obtainable from the simple generalized gradient-diffusion hypothesis (GGDH) hardly align with the simulation results in wall-shear flows at Pr ⩾0.71. However, the GGDH form returns a quite reasonable approximation in shear-free flow regions and/or lower Pr fluid cases. In the former flow cases, it has been found that an introduction of quadratic products of the Reynolds-stress tensor into the gradient diffusion model may improve the predictive performance.

An Algebraic Model of Turbulence for Flows with a Maximum of Tangential Stress Inside the Boundary Layer

The two-layer Klauser scheme of turbulent boundary layer is used to construct a new algebraic model of turbulence for a fairly wide class of flows characterized by the presence of a maximum of tangential stress inside the layer. A distinguishing feature of the model is the method of determination of the velocity scale of turbulence in the outer region of the boundary layer, based on the use of the Bradshow–Ferriss–Atwell (BFA) equation for turbulent stresses. This provides the possibility of taking into account the effect on the scale of both the external factors (longitudinal pressure gradient, injection, suction, transverse curvature of the surface) and the basic mechanisms of transport of turbulent tangential stresses (convection, diffusion, generation, and dissipation). In the final analysis, this defines the rather wide capabilities of the model confirmed by the results of its testing on the basis of experimental data for boundary layers with a positive differential pressure, injection, and transverse curvature of the surface.

Computations of inlet/isolator for SCRAMjet engine

The inlet/isolator is one of the important parts to the SCRAMjet engine. In the present paper, the flow fields within an inlet and isolator of a SCRAMjet engine have been simulated by the Reynolds-Averaged Navier-Stokes equations with the Baldwin-Lomax algebraic turbulence model. The second order H. C. Yee TVD scheme is used for the inviscid flux terms and a central difference considering the character of flow is used for the diffusion terms. The code was used to simulate the flow field of three types of the inlet/isolator models. The results show that the configuration of the internal compression surface of the inlet and the divergent angle of the isolator have a great important effect on the characteristics of the inlet/isolator.

Unsteady Blade-Row Flow Calculations Using a Low-Reynolds-Number Turbulence Model

Acomputational study onunsteady compressiblee owshasbeenperformed by adoptinga low-Reynolds-number k‐!turbulencemodelinconjunctionwithdual-time-steppingandmultigridschemes.Anexplicitfour-stageRunge ‐ Kutta scheme for the Navier ‐Stokes equations and an approximate factorization scheme for the k‐! turbulence model equations are adopted for dual-time stepping. Computational results for unsteady shock-wave boundarylayer interaction in the case of shock buffeting phenomena around a bicircular-arc airfoil are in good agreement with extant experimental data, as well as those for blade surface-pressure distributions in the case of rotor-stator interaction in a turbine stage. Comparisons of the results obtained by the k‐! and algebraic turbulence models indicate that theeffect of turbulence models on the surface-pressure distributions is not so signie cant in the case of rotor-stator interaction as in the case of shock buffeting phenomena, which depend on the extent of the separated region involved. However, based on the results obtained by the k‐! model, the effects of the wake from the stator on the boundary-layer transition over the rotor blade surface can be explained by showing that high-intensity turbulence of the stator wake induces an early transition.

COMPUTATIONAL SIMULATION OF DYNAMIC STALL ON THE NLR 7301 AIRFOIL

The dynamic stall behavior of the supercritical NLR 7301 airfoil is analyzed with a 2-D thin-layer Navier–Stokes code. The code solves the compressible Reynolds-averaged Navier-Stokes equations with an upwind biased numerical scheme in combination with the Baldwin–Lomax or the Baldwin–Barth turbulence models. The effect of boundary layer transition is incorporated using the transition length model of Gostelow et al. The transition onset location is determined with Michel's formula or it can be specified as an input parameter. The two turbulence models yield significantly different steady-state lift coefficients at incidences greater than 8°. The use of the one-equation Baldwin–Barth model together with the Gostelow transition model is found to give substantially better agreement with the experimental data of McCroskeyet al. than the Baldwin–Lomax model. Also, the unsteady computations are strongly affected by the choice of the turbulence model. The Baldwin–Barth model predicts the lift hysteresis loops consistently better than the algebraic turbulence model. However, the one-equation model improves the prediction of the moment hysteresis loops only for one test case.

Numerical Analysis of Turbulent Heat Transfer in a Square Duct with One Rough Wall by Algebraic Turbulent Heat Flux Models.

A numerical analysis has been performed for fully developed turbulent flow and heat transfer in a rectangular duct with smooth and rough walls by using algebraic Reynolds stress and turbulent heat flux models. The wall functions and the universal law of the wall, which are used as the boundary conditions of turbulent energy and dissipation, apply in the present analysis instead of taking shape of roughness element into account. Therefore, the roughness enters through the log law relating the velocity at the first grid point away from the wall with the friction velocity. In this numerical analysis, two kinds of turbulent heat flux models, i. e. Lumley-Launder model and Jones-Musonge model, are examined to compare the contour distributions of turbulent heat fluxes in three dimensions which have been measured in detail. In numerical analysis, convection and diffusion terms for the transport equation of turbulent heat flux are modelled as an algebraic turbulent heat flux model. From the comparison calculated results with the experimental data, it is found that two models can predict the mean temparature distributions which are distorted near the smooth wall located opposite to rough wall side. On top of that, Lumley-Launder model is able to better realize mean temperature near smooth and rough walls than Jones-Musonge model. As for comparison of the turbulent heat flux, calculated results suggest that two models predict characteristic features while two models have tendency to overestimate the experimental values.

An optimised short bearing theory for nonlinear dynamic analysis of turbulent journal bearings

An approximate method for nonlinear dynamic analysis of turbulent journal bearings supporting an unbalanced rigid shaft is proposed. The method is based on two assumptions: the separation of variables for pressure and a parabolic pressure distribution in the axial direction of the bearing. To take into account inertia effects, the well-known algebraic turbulent model based on the Prandtl mixing length hypothesis is used. Using the Constantinescu's approach, the pressure equation is modified by introducing two turbulent coefficients which are depending on the local Reynolds number. The nonlinear equations of motion for the rotor-bearing system are solved by means of the Euler's scheme, and the journal centre trajectories are examined for cases with and without unbalance forces. To illustrate the validity of the present study, three cases of journal bearings are analysed. The accuracy of the minimum film thickness, the dynamic transmissibility coefficient and the peak-to-peak displacement amplitudes obtained by the proposed methodology are comparable to the more elaborate and time consuming 2-D finite difference solution, while the turbulent journal centre orbits are comparable to those obtained experimentally. It is concluded that the optimised short bearing theory shows the advantage of minimising the computation time required for nonlinear dynamic analysis of laminar and turbulent journal bearings without any significant loss of accuracy.

Viscous simulation of F-16 like configuration at high AoA

A viscous simulation of the flow over an F-16A model with inlet and the ventral fins, which were normally not considered in previous studies, is described. The method uses a patch grid multi-block, multi-grid Navier-Stokes solver. A novel method to grid the intake and diverter is used. The arrangement of the gridding system adopted minimises the number of grids needed particularly for viscous computations. A modified form of the Baldwin-Lomax algebraic turbulence model was used for the high AoA studies. To simplify the incorporation of turbulence model, the blocking arrangement is such that it allows a thin-layer Navier-Stokes equation formulation in the calculation of the eddy viscosity. A series of four flow conditions were studied; M = 0.9, AoA = 4°, 6.0°; and M = 1.2 at AoA = 6.0°; and M = 1.85 at AoA = 16°. Extensive comparisons with experimental data of the pressure distribution on the wing and on different streamwise stations show the modelling to be reasonably accurate. For the results presented here, the wing tip missile is ignored.

Fully developed turbulent flow in ducts with symmetric and asymmetric rough walls

A low computational cost semi-analytical procedure is presented for the solution of incompressible fully developed turbulent flow through symmetric and asymmetric ducts with rough walls. Using a modified algebraic turbulence model to represent the influence of rough surfaces, the flow field in ducts with different wall roughnesses is solved. The velocity distribution and the friction factor in two different geometries, circular duct and parallel plates, are presented and compared with experimental and numerical data available in the literature, and a satisfactory agreement is obtained.

Wall boundary layer in a stratified medium

An algebraic model of turbulence in a stratified medium is constructed using equations of second-order moments within the framework of a two-layer scheme.

Numerical Analysis of Turbulent Flow of Non-Newtonian Fluids in Curved Pipes.

A numerical analysis has been performed for developing turbulent flow of non-Newtonian fluid in curved pipe, Numerical results are shown in two cases of power-law fluids, i.e. power index 0.76 and 0.9, at Reynolds number 19000. The ratio of bend mean radius of curvature to radius is 10 and straight duct of 40 and 100 diameters are attached to the inlet and outlet of the bend, respectively. In numerical analysis, an algebraic Reynolds stress model was adopted in order to predict precisely the anisotropic turbulent flow and boundary-fitted coordinate system was introduced as the method of coordinate transformation. The numerical results are compared with the experimental data measured by laser-Doppler anemometer. Mean velocity and fluctuating velocity in axial direction are examined into detail to clarify the validity of the turbulent model and present numerical method. As a result of this research, it is found that the present method could predict well the streamwise mean-velocity in both cases of power-law fluids. As for the comparison of fluctuating velocity, characteristic features are reproduced except for the outer region of pipes. The present method predicts its value in outer regin smaller than that of the experiment, while the present method realized the phenomenon of decaying mean fluctuating velocity with decreasing power index. The results of comparison with the experimental data suggest that algebraic turbulent model is applicable to the non-Nowtonian fluid although agreement between both results is certainly not perfect in all detail.

Calculation of flow of a viscous compressed Gas in the casing of low-flow centrifugal Compressor stages

UDC 621.515.5:533.21.001.24 Calculations of the flow of three-dimensional viscous compressed gas in the casing of low-flow stages of centrifugal compressors (CC) are very laborious. Many problems, for example, determination of the turbulent viscosity of such flow, have not been solved sufficiently accurately and clearly. The complexity of the problem is also due to the nonstationary character of the flow itself. However, even approximate solutions make it possible to refine the physical and mathematical models of the flow and find ways of further improving the design of the casing. A method of calculating the flow with the method of finite differences in radial vaneless slots (vaneless diffusers (VLD), side pockets around the impeller (Im) disks, etc.) used for analyzing the flow in VLD was developed at SPSTU [I]. A comparison of the calculated and experimental data demonstrated the sufficiently high efficiency of the method for use as an algebraic model of turbulence, like the k---e model, and the possibility of taking into consideration the anisotropy of the roughness of the diffuser walls. The distribution of the values of the local loss factor by radius in VLD behind half-open Im for different angles ot 2 at the inlet with constant (5, 10, 15 ~ and also at variable (5-15 ~ and 15-5 ~ values along axis z is shown in Fig. 1. The first maximum is due to restructuring of the flow at the inlet to VLD behind Ira, and the second corresponds to local burbling zones. With a decrease in angle tx2, the values of the local loss factor increase as a result of an increase in the path length for a given Ar. The presence of flanges decreases the losses for small angles ct 2 around it and increases them for constant angle cr 2 or for higher values around it due to warping of the flow. Figure 2 shows the important effect of angle ct 2 and the ratio b 2 = b2/D 2 on the losses in VLD [ 1, 2]. We now propose investigating the flow in the side pockets around Im with the method developed. In the second stage, the analysis of the flow in the casing of a CC stage was conducted together with the Institute of Turbomachines at Hannover University (Germany) with Start CD software by the finite volumes method [3, 4]. The calculation was performed for a series of low-flow closed Ira, including Im of the "channel type," and immobile stage elements [VLD + reverse control apparatuses (RCA)]. The general shape of the region of the flow in the immobile elements is shown in Fig. 3. Note the reciprocal effect of the elements of the rotating elbow (RE) on flow in VLD. The existence of secondary flows is characteristic of RCA. For qb = 0.0188 (significantly lower than the calculated value) on the side of the vacuum, burbled flow arises again at the beginning of RCA and develops along its length due to secondary flows (Fig. 4). In tests of the SVD-6 low-flow "channel" impeller having qbop t -- 0.014, relatively high indexes and a wide zone of stable operation were obtained [4]. For modes close to the calculated mode, flow had a relatively stable character (Fig. 5a), while for low flow rates, developed burbled zones were observed in the channel (Fig. 5b). The third stage of the studies consisted of developing target software for calculating three-dimensional compressed viscous flow in the casing of a CC stage by the method of finite volumes based on the data on turbulent viscosity in [5]. A three-dimensional network sufficiently completely describes the geometry of the casing. The network is constructed with consideration of the effect of internal gas leaks and cross flows. The casing of the stage is divided into two parts: ultralow-flow Im with VLD and immobile elements (part of the VLD, RE, RCA, and ring confusor (RC). The stage casing is divided into two

Investigations of High-Speed Civil Transport Inlet Unstart at Angle of Attack

The unstart transient of a high-speed civil transport mixed compression axisymmetric inlet at Mach 2 and 2-deg angle of attack was investigated numerically by using a three-dimensional time-accurate Navier‐Stokes solver. The Baldwin ‐Lomax algebraic turbulence model and an extrapolation uniform mass bleed boundary condition for the slot bleed were employed. It is observed that, when an angle of attack is imposed, the e ow on the leeward side has a stronger compression than that at zero angle of attack. The strong compression reduces the Mach number upstream of the terminal shock and therefore makes the shock move upstream e rst on the leeward side. The initial shock motion starts with the bifurcation of the terminal shock. The lower part of the split shock is stable because of the centerbody bleed, whereas the top part of the shock continues to travel upstream. When the terminal shock on the leeward side passes the bleed region, a separation is induced by the shock /boundary-layer interaction on the shoulder of the inlet centerbody, and the entire inlet is brought to unstart. The overall computed e owe eld phenomena agree qualitatively with the experimental observations.

Numerical Simulation of High-Speed Civil Transport Inlet Operability with Angle of Attack

A high-speed civil transport inlet at Mach 2 and angle of attack is simulated by using a three-dimensional Navier‐Stokes solver with the Baldwin ‐Lomax algebraic turbulence model. An extrapolation uniform mass bleed boundary condition for the slot bleed is successfully employed. At zero angle of attack and critical operation, the computational pressure distribution agrees well with the experiment. The location and intensity of the terminal shock and the total pressure recovery are accurately computed. The predicted steady-state distortion deviates from the experiment. The computed maximum angle of attack that the inlet can sustain before unstart is in good agreement with the experiment. Mesh ree nement results are presented.

Navier-Stokes analysis of vortex flow over a cropped delta wing

The vortex flow over a 65 deg cropped delta wing with round leading edge, at Mx=0.85 and Rex=2.38 x 10 power 6, has been analyzed for 10 deg, 20 deg, and 30 deg angles of attack. A vertex based finite volume code, JUMB03D, with an algebraic turbulence model has been used to13; solve the Reynolds Averaged Navier Stokes (RAS) equations. An H-O type grid generated by a hybrid elliptio-algebraic method has been used here, and a significant improvement of the results over an O-O type grid has been obtained. The results are compared with available experimental data. The complex physical phenomena due to interactions among the primary, secondary, and tertiary vortices, cross-flow and terminating shocks, and turbulent boundary layer, as visualized from the numerical solution are presented and discussed here.

Developing turbulent duct flow: hybrid solution via integral transforms and algebraic model

The integral transform method is employed for the hybrid numerical‐analytical solution of the boundary layer formulation in developing turbulent flow inside channels. An algebraic turbulence model due to Cebeci and Smith is adopted, in light of its popularity demonstrated in the recent numerical simulation literature. Numerical results for the velocity components in a parallel‐plates channel are obtained, and the automatic error control feature of the present approach is demonstrated. Critical comparisons with experimental works, as well as with previous simulations that employed differential turbulence models, are also performed.

Toward the Development of an Algebraic Turbulent Heat Flux Model with the Aid of LES Data.

In order to derive a possible direction for improvements of an algebraic heat flux model, a priori explorations were attempted by processing the LES data presently performed under various flow and heat transfer conditions including shear-free boundaries. The present calibration has elucidated that the turbulent heat flux vectors obtainable from the simple GGDH hardly align with the simulation results in wall-shear flows with a wall normal temperature gradient of fluids at Prg0.71. However, the GGDH returns a quite reasonable approximation for heat flux vectors in shear-free flow regions and/or lower Pr fluid cases. In the former flow cases, it has been found that a further introduction of a tensor expression, consisting of quadratic products of the Reynolds stress tensor, into the gradient diffusion model may improve the predictive performance.

Attenuating Artificial Dissipation in the Computation of Navier–Stokes Turbulent Boundary Layers

We propose a new formulation of the fourth-difference artificial dissipation coefficient needed for the Navier–Stokes solutions. This coefficient is scaled by a damping function which is expressed in terms of the Baldwin–Lomax algebraic turbulence model. The suggested scaling function damps the artificial dissipation across the boundary layer. The objective of this paper is to test the ability of the suggested damped scaling coefficient to provide (a) a given accuracy on a coarser grids and (b) an accurate computing of turbulent boundary layers. To accomplish this, attached and separated transonic flows over the NACA 0012 airfoil, and turbulent flow over a flat plate have been considered.

Calculations of turbulent jets with finely dispersed solid admixtures

We suggest a computational-experimental method for determining the hydrodynamic characteristics of a dust-laden gas jet. The method is based on Prandtl-Mieses generalized variables. For this case we obtain an algebraic model of turbulence which allows us to take into account the mutual effect of phases and the initial conditions at the nozzle outlet.