Compressible Boundary Layer Flow Research Articles

Analysis and extension of field synergy principle (FSP) for compressible boundary-layer heat transfer

The major purpose of this article is to extend the field synergy principle from incompressible fluid flow to compressible fluid flow. Theoretical analysis and numerical simulation are both conducted. Theoretical analysis indicates that for compressible boundary-layer flow, dot production of static pressure gradient and velocity, and viscous dissipation could be involved in energy equation, which are absent in the incompressible boundary-layer energy equation. Therefore, the heat transfer of the compressible flow is determined by the synergy between total enthalpy gradient and velocity, instead by the synergy between static temperature gradient and velocity, thereby extending the FSP from incompressible fluid flow to compressible flow. Detailed numerical analysis is conducted for the compressible boundary-layer flow to numerically verify the above findings.

Global Instability Analysis of Laminar Boundary Layer Flow Over a Bump at Transonic Conditions

Modal three-dimensional BiGlobal linear instability analysis is performed in steady, spanwise-homogeneous two-dimensional laminar compressible boundary-layer flow past a millimeter-tall hemispherical bump at transonic conditions. Starting with subsonic inlet flow, at the flow conditions considered a stationary shock is formed near the downstream end of the bump. The interplay of shock and adverse-pressure-gradient results in a steady spanwise homogeneous laminar two-dimensional laminar separation bubble being formed at the downstream end of the bump.The objective of the present analysis is to interrogate this basic flow with respect to its potential to sustain low-frequency unsteadiness arising from linear amplification of unstable traveling global flow eigenmodes. Such unsteadiness, coupled to eigen- frequencies of the structure, can lead to resonance phenomena that are detrimental for the performance and adversely affect the efficiency of systems on which the bump configuration is employed. Only damped global eigenmodes have been identified at the parameters examined, pointing to the possibility of the above mentioned unsteadiness being the result of algebraic instability.

Approximate Analytical Solution of Magnetohydrodynamics Compressible Boundary Layer flow with Pressure Gradient and suction/injection

The aim of this work is to obtain exact analytical solution to the two dimensional laminar compressible boundary layer flow with an adverse pressure gradient in the presence of heat and mass transfer with MHD. The method applied is homotopy analysis method. It is shown that this solution agrees very well with numerical solution which is obtained by Runge-Kutta Merson method and results are shown graphically for different magnetic parameters.

Similarity Solution of the Influence of the Thermal Radiation and Heat Transfer on Steady Compressible Boundary Layer Flow

This paper determines the influence of the radiation and heat transfer on the compressible boundary layer flow using similarity solutions approach. The Roseland approximation is used to describe the radiative heat flux in the energy equation and the compressible boundary layer equations are transformed using Stewartson transformation. Similarity (invariant) solutions for the governing partial differential equations system are constructed. The shooting method is employed to transform the resulting non-linear boundary value problem into initial value problem, which is solved numerically. The effects of various parameters on the velocity and temperature profiles as well as the Falkner skan exponent and Prandtl number are shown graphically.

Inner/outer layer interactions in turbulent boundary layers: A refined measure for the large-scale amplitude modulation mechanism

The amplitude modulation (AM) imparted by the outer layer large-scale motions on the near-wall turbulence is studied through direct numerical simulation of compressible boundary layer flow at moderate Reynolds number. Mathis et al. [J. Fluid Mech. 628, 311 (2009)] introduced an amplitude modulation coefficient to quantify this effect, whereby carrier and modulated signals are decoupled through a procedure based on the Hilbert transform of the streamwise velocity signals. However, Schlatter and Örlü [Phys. Fluids 22, 051704 (2010)] have recently shown that a non-zero amplitude modulation coefficient is closely associated to a non-zero value of the velocity skewness, and therefore, it does not necessarily reflect genuine physics. In this paper, the analysis is extended through systematic use of the two-point amplitude modulation correlation, which is shown to be a refined measure of the top-down influence of large-scale outer events on the inner part of the boundary layer.

Generalized conservative approximations of split convective derivative operators

We propose a strategy to design locally conservative finite-difference approximations of convective derivatives for shock-free compressible flows with arbitrary order of accuracy, that generalizes the approach of Ducros et al. (2000) [1], and that can be applied as a building block of low-dissipative, hybrid shock-capturing methods. The approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible Euler equations, which guarantee strong numerical stability and (near) energy preservation in the inviscid limit. A convenient implementation of the high-order fluxes is suggested, which guarantees improved computational efficiency over existing methods. Numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten, and effectiveness of convective splitting of Kennedy and Gruber (2008) [2] in providing extra stability in the presence of strong density variations. Numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non-uniform grids, and overall support superior computational efficiency of high-order schemes.

Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer

The motivation of the present study is to obtain exact analytical solution of the steady laminar flow of a compressible viscous fluid over a rotating disk subjected to a uniformly applied suction or blowing. Classical Von Karman problem of a rotating disk is extended to account for the compressibility effects with insulated and isothermal wall conditions. Using Von Karman similarity transformation the compressible nonlinear equations of motion are reduced to a boundary value problem whose solution was first obtained by Ackroyd [J. A. D. Ackroyd, “On the steady flow produced by a rotating disc with either surface suction of injection,” J. Eng. Phys. 12, 207 (1978)] for the velocity field in terms of a series of exponentially decaying functions. This kind of an approach, however, besides being incapable of resolving the velocity field for higher values of injection (see the conclusion of Ackroyd) is also shown not to be suitable for the temperature distribution of the compressible flow, necessitating the use of a Chebyshev collocation technique in such circumstances. A universally valid analytical technique based on the homotopy analysis method is next applied to obtain solutions corresponding to both velocity and temperature fields. This method yields explicit analytic solutions converging uniformly to the exact solution having the form of exponentially decaying functions for the full range of parameters considered. The effects of suction and blowing together with compressibility on the physically relevant and significant parameters are rigorously explored from the exact formulas extracted.

Compressible Boundary-Layer Predictions at High Reynolds Number Using Hybrid LES/RANS Methods

Simulations of compressible boundary-layer flow at three different Reynolds numbers (Re δ = 5.59 × 10 4 , 1.78 × 10 5 , and 1.58 x 10 6 ) are performed using a hybrid large-eddy simulation/Reynolds-averaged Navier-Stokes method. Variations in the recycling/rescaling method, the higher order extension, the choice of primitive variables, the Reynolds-averaged Navier-Stokes to large eddy simulation transition parameters, and the mesh resolution are considered in order to assess the model. The results indicate that the present model can provide good predictions of the mean-flow properties, second-moment statistics, and structural features of the boundary layers considered. Normalized turbulent statistics in the outer layer are found to be independent of Reynolds number, similar to incompressible turbulent boundary layers.

Modelling flow transition in a hypersonic boundary layer with Reynolds-averaged Navier-Stokes approach

Based on Reynolds-averaged Navier-Stokes approach, a laminar-turbulence transition model is proposed in this study that takes into account the effects of different instability modes associated with the variations in Mach numbers of compressible boundary layer flows. The model is based on k-ω-γ three-equation eddy-viscosity concept with k representing the fluctuating kinetic energy, ω the specific dissipation rate and the intermittency factor γ. The particular features of the model are that: 1) k includes the non-turbulent, as well as turbulent fluctuations; 2) a transport equation for the intermittency factor γ is proposed here with a source term set to trigger the transition onset; 3) through the introduction of a new length scale normal to wall, the present model employs the local variables only avoiding the use of the integral parameters, like the boundary layer thickness δ, which are often cost-ineffective with the modern CFD (Computational Fluid Dynamics) methods; 4) in the fully turbulent region, the model retreats to the well-known k-ω SST (Shear Stress Transport) model. This model is validated with a number of available experiments on boundary layer transitions including the incompressible, supersonic and hypersonic flows past flat plates, straight/flared cones at zero incidences, etc. It is demonstrated that the present model can be successfully applied to the engineering calculations of a variety of aerodynamic flow transition.

Influence of suction/blowing on the finite amplitude disturbances having non-stationary modes of a compressible boundary layer flow

In this paper, a weakly non-linear stability analysis is pursued to explore the effects of suction and blowing on a compressible mode of instability of the three-dimensional boundary layer flow induced by a rotating disk. The main thrust of the research is to extend the stationary work of Seddougui and Bassom (1996) to cover for the non-stationary mode so that it is targeted to determine whether the major role in finite amplitude destabilization of the boundary layer is played by the stationary mode of Seddougui and Bassom (1996) or the non-stationary mode as calculated from the present study. Within this perspective, the basic compressible flow obtained in the large Reynolds number limit, together with a suction parameter entering into the wall with normal velocity at the wall, is perturbed by disturbances which are constituted as the non-linear interaction of fundamental modes and harmonics. The effects of non-linearity are then explored by deriving a finite amplitude equation governing the evolution of the non-linear lower branch modes and also allowing the finite amplitude growth of a disturbance close to the neutral location determined from a prior stability analysis. Although the form of the amplitude equation is not at all surprising (given a similar result for the stationary vortices in Seddougui and Bassom (1996)), the dependence of the coefficients of the Landau-type modulated vortex amplitude equation on the frequency parameter is established here. A close examination of the coefficients of this evolution equation indicates strongly that the non-linearity has a destabilizing effect for both suction and blowing through the surface of the disk, the effect of which is much more stronger for a suction. Also the impact of non-linearity is higher for the non-stationary compressible modes than for the stationary waves of Seddougui (1990) and Seddougui and Bassom (1996). Moreover, in the case of suction, there occurs a regime of non-stationary modes that cover not only the positive frequency waves, but also waves having negative frequencies, and these modes are always unstable no matter whether the wall is insulated or isothermal. The solution of the asymptotic amplitude equation further demonstrates that as the local Mach number increases, compressibility has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable whenever fluid injection is applied. Unlike this, suction makes the underlying flow more convectively unstable as far as the compressibility is concerned, particulary for the modes generated as a consequence of an isothermal wall. Both for the suction and injection cases, disturbances having positive frequency are always shown to cause an instantaneous non-linear amplification prior to the negative frequency waves.

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three‐dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [1] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.

Non-uniform mass transfer or wall enthalpy into a compressible flow over a rotating sphere

A study for the effect of non-uniform slot injection/suction and non-uniform total enthalpy at the wall for a steady non-similar laminar compressible boundary layer flow over a rotating sphere is presented. The effects of rotation and free stream Mach number have also been considered. Non-similar solutions have been obtained from the starting point of the stream wise coordinate to the point of separation using an implicit finite difference scheme in combination with the quasi-linearization technique. The results indicate that the separation can be delayed by non-uniform slot suction and also by moving the slot downstream but the non-uniform slot injection has just the opposite effect. An increase of rotation and total enthalpy at the wall causes separation to occur earlier while cooling delays it. An increase of the Mach number shifts the point of separation upstream due to the adverse pressure gradient. The non-uniform total enthalpy at the wall has a very little effect on the skin friction and hence on the point of separation.

Direct Spatial Resonance in the Compressible Boundary Layer on a Rotating-disk

This paper focuses on a resonance mechanism that can lead to significant disturbance amplification at conditions which are sub-critical to nonlinear instabilities. Particularly, direct spatial resonance instability is investigated, which is present in the basic three-dimensional viscous compressible boundary-layer flow due to a rotating-disk. Within this purpose, the linearized system of stability equations is treated numerically making use of a spectral Chebyshev collocation method. The analysis provides critical resonant Reynolds numbers above which growth occurs. Amplitudes of the response of the degeneracies decaying rapidly due to their high damping rates are shown to exist for small enough Reynolds numbers while the flow is still in the laminar state. If the flow is restricted to the incompressible case, the results of Turkyilmazoglu and Gajjar (in Sadhana Acad P Engs 25:601–617, 2000) are completely reproduced. The influences of compressibility are then explored by means of varying the Mach and Prandtl numbers in the cases of heating/cooling the wall as well as the isothermal wall. In general, compressibility effects are found strongly in favor of stabilizing as the Mach number increases, while a strong destabilization is observed by lowering the critical values of Reynolds numbers in the cases of wall heating and insulation. The modal interaction and coalescence of the eigenmodes calculated here create local algebraic growth by rapid development of relatively large amplitudes which might then provide the onset of nonlinear effects followed by transition.

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three‐dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [1], it was demonstrated that the short‐wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [2] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameter , which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [1] for zero‐suction compressible modes.The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.

Initial-value problem for three-dimensional disturbances in a compressible boundary layer

An initial-value problem is formulated for a three-dimensional wave packet in a compressible boundary layer flow. The problem is solved using a Laplace transform with respect to time and Fourier transforms with respect to the streamwise and spanwise coordinates. The solution can be presented as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. Two discrete modes, known as mode S and mode F, are of interest in high-speed flows since they may be involved in a laminar-turbulent transition scenario. The continuous and discrete spectrum are analyzed numerically for a hypersonic flow with Mach number M=5.6. The following features are revealed: (1) The synchronism of mode S with acoustic waves at a streamwise wave number α→0 is primarily two-dimensional; (2) at high angles of disturbance propagation, mode F is no longer synchronized with entropy and vorticity waves; (3) at high angles of disturbance propagation, the synchronism between mode S and mode F is not accompanied by a mode S instability, and at even higher angles of disturbance propagation, mode S and mode F are not synchronized.

MHD compressible turbulent boundary-layer flow with adverse pressure gradient

The effects of the magnetic field and localized suction on the steady turbulent compressible boundary-layer flow with adverse pressure gradient are numerically studied. The magnetic field is constant and applied transversely to the direction of the flow (global or local). The fluid flow is subjected to a constant velocity of localized suction, and there is no heat transfer between the fluid and the plate (adiabatic plate). The Reynolds-Averaged Boundary-Layer (RABL) equations and their boundary conditions are transformed using the compressible Falkner-Skan transformation. The resulting coupled and nonlinear system of PDEs is solved using the Keller’s box method. For the eddy-kinematic viscosity the turbulent models of Cebeci-Smith and Baldwin-Lomax are employed. For the turbulent Prandtl number the extended Kays-Crawford’s model is used. The flow is subjected to an adverse pressure gradient. The obtained results show that the flow field can be controlled by the applied magnetic field as well as by localized suction.

Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry

An analysis has been performed to study the unsteady laminar compressible boundary layer governing the hypersonic flow over a circular cone at an angle of attack near a plane of symmetry with either inflow or outflow in the presence of suction. The flow is assumed to be steady at time t=0 and at t>0 it becomes unsteady due to the time-dependent free stream velocity which varies arbitrarily with time. The nonlinear coupled parabolic partial differential equations under boundary layer approximations have been solved by using an implicit finite-difference method. It is found that suction plays an important role in stabilising the fluid motion and in obtaining unique solution of the problem. The effect of the cross flow parameter is found to be more pronounced on the cross flow surface shear stress than on the streamwise surface shear stress and surface heat transfer. Beyond a certain value of the cross flow parameter overshoot in the cross flow velocity occurs and the magnitude of this overshoot increases with the cross flow parameter. The time variation of the streamwise surface shear stress is more significant than that of the cross flow surface shear stress and surface heat transfer. The suction and the total enthalpy at the wall exert strong influence on the streamwise and cross flow surface shear stresses and the surface heat transfer except that the effect of suction on the cross flow surface shear stress is small.

Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating‐Disk

In this article, a theoretical study is pursued to investigate the structure of the lower branch neutral stability modes of three‐dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating‐disk. Special attention is focused on to the short‐wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple‐deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [1] for the incompressible stationary mode and in [2] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. The asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.

A minimal composite theory for stability of non-parallel compressible boundary-layer flow

A new “minimal composite” theory that extends the approach of Govindarajan and Narasimha [1] is proposed here for 2D non-parallel compressible boundary-layer stability subject to 3D disturbances. The mean profiles are obtained from Horton’s analysis, which provides a good approximation for a large range of Prandtl numbers at non-zero pressure gradients. In the lowest order, all effects of order lower than O(R-2/3) anywhere in the boundary-layer are included, R being the local boundary-layer Reynolds number; the resulting non-parallel formulation yields a set of four ordinary differential equations, as compared to the five coupled equations of classical parallel flow theory of Mack [2]. The largest effect on stability of flow non-parallelism is found to be due to the wall-normal advection of velocity and temperature disturbance quantities by the mean flow. The present theory shows that the bulk viscosity, invariably included in compressible stability theories, is irrelevant at the lowest order. In comparison with the “full” [O(R-1)] non-parallel theory, the present theory is marginally better than the parallel flow theory.

Non-uniform multiple slot injection (suction) or wall enthalpy into a compressible flow over a yawed circular cylinder

Steady non-similar laminar compressible boundary layer flows over yawed infinite cylinder with non-uniform multiple slot injection/suction and non-uniform wall enthalpy have been studied. The numerical difficulties are overcome by applying the method of quasi-linear implicit finite difference scheme with an appropriate selection of finer stepsize along the streamwise direction. Separation can be delayed more effectively by non-uniform multiple slot suction and also by moving the slots downstream as compared to the non-uniform single slot suction but the non-uniform multiple slot injection has the reverse effect. Increase of Mach number shifts the point of separation upstream due to the increase in the adverse pressure gradient. An increase of enthalpy at the wall causes separation to occur earlier while cooling delays it. Non-uniform total enthalpy at the wall has very little effect on the skin frictions or the point of separation. Also, the yaw angle has very little effect on the location of the point of separation.