Free-stream Velocity Distributions Research Articles

Evaluation of Turbulence Models in Unsteady Separation

Unsteady separation is a phenomenon that occurs in many flows and results in increased drag, decreased lift, noise emission, and loss of efficiency or failure in flow devices. Turbulence models for the steady or unsteady Reynolds-averaged Navier–Stokes equations (RANS and URANS, respectively) are commonly used in industry; however, their performance is often unsatisfactory. The comparison of RANS results with experimental data does not clearly isolate the modeling errors, since differences with the data may be due to a combination of modeling and numerical errors, and also to possible differences in the boundary conditions. In the present study, we use high-fidelity large-eddy simulation (LES) results to carry out a consistent evaluation of the turbulence models. By using the same numerical scheme and boundary conditions as the LES, and a grid on which grid convergence was achieved, we can isolate modeling errors. The calculations (both LES and RANS) are carried out using a well-validated, second-order-accurate code. Separation is generated by imposing a freestream velocity distribution, that is modulated in time. We examined three frequencies (a rapid, flutter-like oscillation, an intermediate one in which the forcing and the flow have the same timescales, and a quasi-steady one). We also considered three different pressure distributions, one with alternating favorable and adverse pressure gradients (FPGs and APGs, respectively), one oscillating between an APG and a zero-pressure gradient (ZPG), and one with an oscillating APG. All turbulence models capture the general features of this complex unsteady flow as well or better than in similar steady cases. The presence, during the cycle, of times in which the freestream pressure-gradient is close to zero affects significantly the model performance. Comparing our results with those in the literature indicates that numerical errors due to the type of discretization and the grid resolution are as significant as those due to the turbulence model.

Comparison of smooth- and rough-wall non-equilibrium boundary layers with favourable and adverse pressure gradients

Measurements were made in rough-wall boundary layers subject to favourable, zero and adverse pressure gradients. Profiles of mean velocity and turbulence quantities were acquired and velocity fields were measured in multiple planes to document flow structure. Comparisons were made to equivalent smooth-wall cases with the same free stream velocity distributions. Outer layer similarity was observed between the rough- and smooth-wall cases in all quantities in the favourable and zero pressure gradient regions, but large differences were observed with adverse pressure gradients. In both the smooth- and rough-wall cases, the favourable pressure gradient reduced the turbulence in the boundary layer, and increased the size of turbulence structures relative to the boundary layer thickness in both the streamwise and spanwise directions, while lowering their inclination angle with respect to the wall. When the boundary layer was returned to a zero pressure gradient following the favourable pressure gradient region, the turbulence level and the size and inclination of the structures returned to their canonical zero pressure gradient condition. The response of the boundary layer was somewhat faster in the rough-wall case, causing it to reach equilibrium in a shorter streamwise distance after the changes in pressure gradient than in the smooth-wall case. The adverse pressure gradient increased turbulence levels relative to the wall friction velocity, reduced the size of turbulence structures relative to the boundary layer thickness and increased their inclination angle. The changes with the adverse pressure gradient were significantly larger with the rough wall than the smooth. The results suggest that similarity might be achieved with adverse pressure gradients if smooth- and rough-wall cases with the same Clauser pressure gradient parameter history are compared.

Instability and Transition of a Boundary Layer over a Backward-Facing Step

The development of secondary instabilities in a boundary layer over a backward-facing step is investigated numerically. Two step heights are considered, h/δo*=0.5 and 1.0 (where δo* is the displacement thickness at the step location), in addition to a reference flat-plate case. A case with a realistic freestream-velocity distribution is also examined. A controlled K-type transition is initiated using a narrow ribbon upstream of the step, which generates small and monochromatic perturbations by periodic blowing and suction. A well-resolved direct numerical simulation is performed. The step height and the imposed freestream-velocity distribution exert a significant influence on the transition process. The results for the h/δo*=1.0 case exhibit a rapid transition primarily due to the Kelvin–Helmholtz instability downstream of step; non-linear interactions already occur within the recirculation region, and the initial symmetry and periodicity of the flow are lost by the middle stage of transition. In contrast, case h/δo*=0.5 presents a transition road map in which transition occurs far downstream of the step, and the flow remains spatially symmetric and temporally periodic until the late stage of transition. A realistic freestream-velocity distribution (which induces an adverse pressure gradient) advances the onset of transition to turbulence, but does not fundamentally modify the flow features observed in the zero-pressure gradient case. Considering the budgets of the perturbation kinetic energy, both the step and the induced pressure-gradient increase, rather than modify, the energy transfer.

Laser-induced fluorescence velocimetry for a hypersonic leading-edge separation

Two-dimensional mapping of the velocity distribution for a hypersonic leading-edge separation flowfield generated by a “tick” shaped geometry is presented for the first time. Discrete measurements of two velocity components were acquired at a flow condition having a total specific enthalpy of 3.8 MJ/kg by imaging nitric oxide fluorescence over numerous runs of the hypersonic tunnel at the Australian Defence Force Academy (T-ADFA). The measured freestream velocity distribution exhibited some non-uniformity, which is hypothesized to originate from images acquired using a set of ultraviolet specific mirrors mounted on the shock tunnel deflecting under load during a run of the facility, slightly changing the laser sheet orientation. The flow separation point was measured to occur at 1.4 ± 0.2 mm from the model leading edge, based on the origin of the free shear layer emanating from the expansion surface. Reattachment of this free shear layer on the compression surface occurred at 59.0 ± 0.2 mm from the model vertex. Recirculating the flow bound by the separation and reattachment points contained supersonic reverse flow and areas of subsonic flow aligned with the location of three identified counter-rotating vortices. A comparison of the recirculation flow streamline plots with those computed using Navier–Stokes and direct simulation Monte Carlo (DSMC) codes showed differences in flow structures. At a flow time close to that produced by the facility, flow structures generated by the DSMC solution were seen to agree more favorably with the experiment than those generated by the Navier–Stokes solver due to its ability to better characterize separation by modeling the strong viscous interactions and rarefaction at the leading edge. The primary reason for this is that the no-slip condition used in the Navier–Stokes solution predicts a closer separation point to the leading edge and structures when compared to the DSMC solution, which affects surface shear stress and heat flux, leading to a difference in flow structures downstream of the separation.

Large-eddy simulations of adverse pressure gradient turbulent boundary layers

Adverse pressure-gradient (APG) turbulent boundary layers (TBL) are studied by performing well-resolved large-eddy simulations. The pressure gradient is imposed by defining the free-stream velocity distribution with the description of a power law. Different inflow conditions, box sizes and upper boundary conditions are tested in order to determine the final set-up. The statistics of turbulent boundary layers with two different power-law coefficients and thus magnitudes of adverse pressure gradients are then compared to zero pressure-gradient (ZPG) data. The effect of the APG on TBLs is manifested in the mean flow through a much more prominent wake region and in the Reynolds stresses through the existence of an outer peak. The pre-multiplied energy budgets show that more energy is transported from the near-wall region to farther away from the wall.

Generaleristics of unsteady MHD temperature boundary layer

Unsteady temperature magnetohydrodynamic (MHD) boundary layer on the arbitrary shape body which temperature is function of time and longitudinal coordinate has been studied. A generalized similarity method is applied to solve the governing non-linear differential equations. Universal partial differential equations of the discussed problem are obtained using the method by which the equations are transformed in such a way that neither the equations nor the boundary conditions depended on particular data of the considered problem. These equations are solved numerically and the integration results are given for different values of introduced parameters. Those solutions permit, before their application to the particular cases (assigned free stream velocity), a generalized analysis of the effects of certain pertinent parameters to the development of the boundary layer. Obtained results are discussed in detail for different values of introduced parameters. Based on the given universal solutions result it is possible to calculate any particular problem for defined free stream velocity and body temperature distribution.

Unsteady MHD stagnation-point flow with heat and mass transfer for a three-dimensional porous body in the presence of heat generation/absorption and chemical reaction

The problem of unsteady Magnetohydrodynamic (MHD) flow with heat and mass transfer near the stagnation point of a three-dimensional (3D) porous body in the presence of heat source/sink and chemical reaction effect has been studied numerically using an efficient iterative implicit finite-difference method. The numerical results are validated by favourable comparisons with previously published work. Three forms for the free stream velocity distributions namely a constantly accelerating flow, a periodic fluctuating flow and an exponentially decelerating flow are considered. Numerical results for the velocity components in the x-and y-directions, temperature distribution and concentration distribution as well as the skin-friction coefficients and the Nusselt and Sherwood numbers are presented graphically for various parametric conditions and discussed.

Simulation of the effect of high-intensity turbulence flow parameters on unsteady boundary layers with streamwise pressure gradients

The dynamic and thermal characteristics of steady and unsteady wall boundary layers are investigated on the basis of the two-parameter turbulence model under conditions of high-intensity free-stream turbulence and exposure to a negative streamwise pressure gradient. Flows from the laminar to the completely developed turbulent regimes, including the transition regime, are calculated. The possibility of the theoretical description of the transition and the developed turbulent regime by means of a quasi-steady-state turbulence model when the free-stream velocity distributions are time-periodic is demonstrated. The properties of the calculated characteristics of the dynamic and thermal boundary layers are analyzed for various oscillation parameters and a moderate steamwise pressure gradient. The numerical results are compared with experimental and theoretical data.

High Reynolds-number orographic wave-breaking experiments

In this paper, we present laboratory measurements at high Reynolds numbers of the turbulence generated by breaking internal-gravity waves of flow over Gaussian-shaped obstacles. The results confirm that the flow evolution leading to wave-breaking for a flow started from rest and the ensuing large-scale flow structures in the wave-breaking region are essentially the same as observed in the low Reynolds-number laboratory experiments of Eiff and Bonneton [Eiff, O.S., Bonneton P., 2000. Lee-wave breaking over obstacles in stratified flow. Phys. Fluids 12 (6),1073–1086] and the associated direct numerical simulations of Gheusi et al. [Gheusi, F., Stein J., Eiff O.S., 2000. A numerical study of three-dimensional orographic gravity-wave breaking observed in a hydraulic tank. J. Fluid Mech. 410, 67–99]. The flow reaches a quasi-steady state during which turbulence statistics are computed in a horizontal plane intersecting the wave-breaking region. The statistics reveal a highly turbulent flow with intensities up to 20% of the free-stream velocity and spatial distributions which indicate that the underlying large-scale flow structures – toroidal vortices – are quasi-steadily aligned in the spanwise direction and that they might be shed downstream.

Computational Fluid Dynamics Study of Wake-Induced Transition on a Compressor-Like Flat Plate

Recent experimental work has documented the importance of wake passing on the behavior of transitional boundary layers on the suction surface of axial compressor blades. This paper documents computational fluid dynamics (CFD) simulations using a commercially available general-purpose CFD solver, performed on a representative case with unsteady transitional behavior. The study implements an advanced version of a three-equation eddy-viscosity model previously developed and documented by the authors, which is capable of resolving boundary layer transition. It is applied to the test cases of steady and unsteady boundary layer transition on a two-dimensional flat plate geometry with a freestream velocity distribution representative of the suction side of a compressor airfoil. The CFD results are analyzed and compared to a similar experimental test case from the open literature. Results with the model show a dramatic improvement over more typical Reynolds-averaged Navier–Stokes (RANS)-based modeling approaches, and highlight the importance of resolving transition in both steady and unsteady compressor aerosimulations.

The effect of negative shear on the transitional separated flow around a semi-circular leading edge

The effect of a negative free-stream mean-shear velocity distribution on the boundary layer development on a flat plate with a semi-circular leading edge is studied experimentally and computationally. The geometry is the same as in the T3L test case of the ERCOFTAC Special Interest Group on Transition. The existence of a negative shear is related to the transition of the boundary layer from laminar to turbulent through separation. The flow investigated here has the same general characteristics as the one presented in a recent work by the authors, Palikaras et al. [Int. J. Heat Fluid Flow 23 (2002) 455–470], where the boundary layer development has been studied under free-stream conditions of uniform and positive mean-shear velocity distributions. The negative shear flow in the core region of the wind tunnel has a value ∂ U/ ∂y=−27.7 s −1, which is the opposite to the case examined by Palikaras et al. [Int. J. Heat Fluid Flow 23 (2002) 455–470]. In the first part of the paper, a detailed description of the flow is given. The measured quantities are presented, discussed and compared with the computational analysis in order to obtain a complete picture of the investigated flow. For the computations, the non-linear k– ε model of Craft et al. [Int. J. Heat Fluid Flow 17 (1996) 108–115] is used, and satisfactory predictions are reported. In the second part, a detailed comparison of the results with the cases of uniform and positive mean-shear velocity inlet distribution is carried out. In the case of negative mean velocity the separated boundary layer leads to a larger reverse flow region than the two other cases. A relation is observed between the location of the stagnation point at the leading edge and the presence or absence of shear. When mean shear is present, depending on the sign, there is a movement of the stagnation point away from the symmetry line of the flat plate and it is believed that this is the driving mechanism that affects the boundary layer development and the longitudinal size of the reverse flow region. This remark is supported by the observation that for all the three cases studied, the longitudinal RMS distribution above the reverse flow region and in the free-stream region has the same values.

Simulation of the Effect of the Freestream Turbulence Parameters on Heat Transfer in an Unsteady Boundary Layer

A variant of the two-parameter turbulence model which makes it possible continuously to calculate a flow region with laminar, transition and turbulent regimes is proposed for investigating the flow under conditions of high freestream turbulence intensity. It is shown that the properties of the thermal transition can be theoretically described using the quasi-steady turbulence model in the case of periodic freestream velocity distribution. The numerical results are compared with theoretical and experimental data. The approach proposed is developed for determining the combined effect of the parameters of harmonic fluctuations of the external velocity and freestream turbulence on the heat transfer characteristics on a flat plate with different boundary conditions for the enthalpy.

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.

Transition on a flat plate with a semi-circular leading edge under uniform and positive shear free-stream flow

The effect of the free-stream velocity profile on the transition from laminar to turbulent flow on a flat plate was studied experimentally and numerically and it is presented in this paper. The flows investigated are based on the T3L test case of the ERCOFTAC Special Investigation Group for transition. According to this test case, the boundary layer development on a flat plate with a semi-circular leading edge is examined by means of transition due to separation, under various free-stream conditions concerning the turbulence intensity and velocity magnitude. In the present work, two free-stream velocity distributions were studied. The first was a uniform velocity one and the second, with a mean shear velocity profile with positive gradient, ∂ U/ ∂y=27.7 s −1. Measurements using hot-wire anemometry were taken in two primary regions: far upstream of the flat plate to observe the velocity and turbulence distributions and near the flat plate to capture the boundary layer development and the transition phenomenon. The effect of the two free-stream velocity distributions was studied and it was shown that for both velocity distributions a recirculation region of the flow occurred near the flat plate wall that led to transition dominated by the boundary layer separation. For the positive velocity gradient the separation region was smaller compared to the case of uniform free-stream profile. Both cases were also studied computationally. Two widely used linear eddy-viscosity turbulence models, the k– ε and the k– ω with specific low Reynolds formulations were applied and in addition, a non-linear eddy-viscosity based on the k– ε model has been implemented. In general, all the k– ε models gave satisfactory predictions for both flow cases regarding the predicted velocity distributions, while the k– ω model gave poor results. Concerning the longitudinal Reynolds stress distributions in the near-wall region, the non-linear k– ε model gave the best predictions inside the separation zone but it over predicted the corresponding values beyond the reattachment point while beyond the separation the linear models predicted the longitudinal stresses in a more satisfactory way.

Unsteady Rotating Flow of a Compressible Fluid over a Finite Disk

The semi-similar formulation of the problem of an unsteady boundary layer over a stationary disk of finite radius induced by a time dependent rotating flow of compressible fluid is considered. The free stream swirling flow is assumed to be tangential and varies as a power of radial distance, i.e., ve ∼︁ r—nϕ(t*), where ϕ(t*) is an arbitrary function of time t*. Several free stream velocity distributions have been examined. Near the outer edge of the disk the flow is described by the Stewartson edge similarity solutions and near the axis of rotation the solutions coincide with the Bödewadt's type of similarity solution. The velocity profiles are monotonic over the outer half of the finite disk but they exhibit oscillation over the inner half (for n < 1). Due to reversal in radial velocity over the inner half of the disk, the boundary layer equations become a kind of time dependent singular parabolic equation and the solution of which requires the conditions over all of its boundaries. Finite-difference scheme using space-centered differences in the axial direction and upwind differences in the radial direction is used for solving the governing equations. Crank-Nicolson scheme is used for time-wise discretisation. Numerical solutions have been obtained for the entire range of the radial distance. Solutions for the entire range of the radial distance exist upto a certain critical value of n. Boundary layer solutions do not exit over the inner half of the disk for the case of potential vortex flow (n = 1). Effect of acceleration and oscillation of the free stream azimuthal velocity on the boundary layer flow and heat transfer have been investigated.

Solving Laminar Boundary Layers as a Sequence of Falkner—Skan Flows

A method for predicting the momentum thickness and skin friction coefficient of laminar boundary layers for arbitrary free–stream velocity distributions has been formulated. The method combines the Thwaites–Walz and Falkner–Skan methods with a numerical technique. The method is simple enough for undergraduate use. It is shown to be accurate when using only a few intermediate points up to the point of separation, making the method suitable for hand-calculation. The method also allows the integration of the Falkner–Skan equation at any particular station to obtain the velocity profile through the boundary layer.

Exact Solution of Nonsteady Thermal Boundary Layer Equation

There are only a few exact solutions of the thermal boundary layer equation. Most of them are derived for a specific surface temperature distribution. The first exact solution of the steady-state boundary layer equation was given for a plate with constant surface temperature and free-stream velocity. The same problem for a plate with polynomial surface temperature distribution was solved by Chapmen and Rubesin. Levy gave the exact solution for the case of a power law distribution of both surface temperature and free-stream velocity. The exact solution of the steady-state boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution was given by the author in two forms: of series and of the integral with an influence function of unheated zone. A similar solution of the nonsteady thermal boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution is presented here. In this case, the coefficients of series depend on time, and in the limit t {r_arrow} {infinity} they become the constant coefficients of a similar solution published before. This solution, unlike the one presented here, does not satisfy the initial conditions at t = 0, and, hence, can be usedmore » only in time after the beginning of the process. The solution in the form of a series becomes a closed-form exact solution for polynomial surface temperature and a power law free-stream velocity distribution. 7 refs., 2 figs.« less

Unsteady nonsimilar two-dimensional and axisymmetric water boundary layers with variable viscosity and prandtl number

The influence of temperature-dependent viscosity and Prandtl number on the unsteady laminar nonsimilar forced convection flow over two-dimensional and axisymmetric bodies has been examined where the unsteadiness and (or) nonsimilarity are (is) due to the free stream velocity, mass transfer, and transverse curvature. The partial differential equations governing the flow which involve three independent variables have been solved numerically using an implicit finite-difference scheme along with a quasilinearization technique. It is found that both the skin friction and heat transfer strongly respond to the unsteady free stream velocity distributions. The unsteadiness and injection cause the location of zero skin friction to move upstream. However, the effect of variable viscosity and Prandtl number is to move it downstream. The heat transfer is found to depend strongly on viscous dissipation, but the skin friction is little affected by it. In general, the results pertaining to variable fluid properties differ significantly, from those of constant fluid properties.

Simultaneous heat and mass transfer under unsteady mixed convection along a vertical slender cylinder embedded in a porous medium

Unsteady laminar mixed convection flow (combined free and forced convection flow) along a vertical slender cylinder embedded in a porous medium under the combined buoyancy effect of thermal and species diffusion has been studied. The effect of the permeability of the medium as well as the magnetic field has been included in the analysis. The partial differential equations with three independent variables governing the flow have been solved numerically using a implicit finite difference scheme in combination with the quasilinearization technique. Computations have been carried out for accelerating, decelerating and oscillatory free stream velocity distributions. The effects of the permeability of the medium, buoyancy forces, transverse curvature and magnetic field on skin friction, heat transfer and mass transfer have been studied. It is found that the effect of free stream velocity distribution is more pronounced on the skin friction than on the heat and mass transfer. The permeability and magnetic parameters increase the skin friction, but reduce the heat and mass transfer. The skin friction, heat transfer and mass transfer are enhanced due to the buoyancy forces and curvature parameter. The heat transfer is strongly dependent on the viscous dissipation parameter and the Prandtl number, and the mass transfer on the Schmidt number.

Unsteady swirling flow of a compressible fluid over an infinite disk with a magnetic field or suction

The laminar boundary layer over a fixed infinite disk induced by a time-dependent vortex flow of compressible fluid is considered. The free-stream velocity has been taken as tangential and varies as a power of radius, i.e.ν ∞ r − n φ(τ) where φ(τ) is an arbitrary function of time t. Several free-stream velocity distributions have been examined. The effect of a magnetic field and suction are also included in the analysis. The solutions are studied for various values of n in the range (− 1 ⩽ n⩽ 1) and also for several values of disk-to-fluid temperature ratio. The oscillations in the velocity profiles decrease with time in the absence of a magnetic field and suction for an accelerating free stream. Heat transfer at the wall is affected slightly by the time variation of free-stream velocity. Solution is possible only for a certain range of n unless a magnetic field or suction is applied.