Vectored injection into compressible laminar and turbulent boundary layers

A fundamental analysis of two-dimensional supersonic boundary layer flow, both laminar and turbulent, is presented for a wide range of normal and nonnormal mass-transfer velocities. The analysis is based on the numerical solution of the Navier-Stokes equations, and results are compared with available theoretical and experimental data. Certain cases of practical importance, for which results are not presently available, are referred to.     

Chebyshev solution of laminar boundary layer flow

An expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method [1], which yields more accurate results than those computed by P. M. Beckett [2] and A. R. Wadia and F. R. Payne [6] as indicated from solving the Falkner-Skan equation, which uses a boundary value technique. This method is accomplished by starting with Chebyshev approximation for the highest-order derivative and generating approximations to the lower-order derivatives through integration of the highest-order derivative.     

The numerical integration of laminar boundary layer equations

Self-similar solutions of boundary layer equations obey non-linear differential equations, automorphic under certain continuous transformation groups. Changes of variables suggested by the theory of continuous LIE groups may reduce the problem to the integration of a first order non linear differential equation, followed by quadratures, thereby greatly simplifying computer integration.The famous Blasius equation, governing the asymptotic laminar boundary layer flow over a semi-infinite plate is presented as a typical example.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Numerical analysis of unsteady compressible laminar boundary layer flow

The unsteady compressible laminar boundary layer flow on an arbitrary cylinder due to an incident stream whose velocity varies arbitrarily with time is considered. The method presented is based on the separation between the convective and diffusive quantities. By defining some new variables, the splitting appears rather naturally, and the initialisation problem can be solved without difficulty. The transformed equations are solved with the help of a semi-implicit finite difference scheme which is unconditionally linearly stable. The computations have been applied to flows past a cylinder with constant and fluctuating free-stream velocities.     

The investigation of a compressible laminar boundary layer past an elliptical cone

The self-similar boundary layer on a sharp circular cone was calculated first by Vvedenskaya[1]. The boundary layer equations were solved in a plane containing an outflow line which lies in this case in the symmetry plane at the leeward side, and after that a solution was constructed by using a marching method along a circumferential coordinate. The calculation results for the boundary layer on an elliptic cone were presented in Bashkin's papers [2–4]. However, only the middle angles of attack (30–50°) were considered. where the outflow line of an external stream is located in the windward symmetry plane, and the flow pattern in the boundary layer is analogous to that of the circular cone. In the present the laminar boundary layer on an elliptic cone is studied for a wide range of angles of attack. The boundary layer has been calculated at small incidence when the outflow of an external flow were located out of the symmetry plane. In this case the equations are solved first in the plane containing the outflow line and then the solutions were constructed by a marching method along a circumferentialcoordinate to the windward and leeward symmetry planes. The distribution of the skin-friction coefficients and the Stanton's numbers on a cone surface was given. The similarity solution of a set of boundary layer equations was obtained for thin cones at large incidence when the stream on the windward side of the cone was directed to the cone nose. The calculations of the laminar boundary layer at hypersonic velocities were carried out to include the real equilibrium properties of the air.     

Accuracy and convergence of a finite element algorithm for laminar boundary layer flow

The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equation appears extensible to the non-linear equations characteristic of laminar boundary layer flow.     

Global PNS solutions for subsonic strong interaction flow over a cone-cylinder-boattail configuration

The solution of the semi-elliptic or so-called parabolized Navier-Stokes equations is considered for large Reynolds numbers and subsonic flows with strong pressure interaction. Flow past a cone-cylinder-boattail configuration is investigated as a prototype of strong viscous-inviscid interaction. A global boundary-layer relaxation procedure is utilized for the formulation of the discrete boundary-value problem. The resulting marching procedure does not require a sub-layer type of approximation. Furthermore, the method does not restrict the step size in the marching direction and is free from any departure effects. Solutions with large recirculation regions are calculated.     

LES of synthetic jets in boundary layer with laminar separation caused by adverse pressure gradient

In the development of synthetic jet actuators (SJAs) for active flow control, numerical simulation has played an important role. In controlling the boundary layer flow separation, an integrated numerical model which includes both the baseline flow and the SJA is still in its initial stage of development. This paper reports preliminary results of simulating the interaction between a synthetic jet and a laminar separation bubble caused by adverse pressure gradient in a boundary layer. The computational domain was three-dimensional and Large-eddy simulation (LES) was adopted. The initial and boundary conditions were defined using or referring to our wind tunnel experimental results. Prior to numerically simulating the interaction between the synthetic jets and the baseline flow, a numerical model for simulating the separation bubble was developed and verified. In the numerical model including the SJA, the synthetic jet velocity at the exit of the SJA was defined as an input. The numerical model was further verified by comparing the simulation with experimental results. Based on reasonable agreement between the numerical and experimental results, simulations were carried out to investigate the dependency of flow control using synthetic jets on the forcing frequency, focused on the lower frequency range of the Tollmien-Schlichting (T-S) instability, and on the forcing amplitude which was represented by the maximum jet velocity at the exit of the SJA. Supporting the hypothesis based on the experiment, LES results showed that the forcing frequency had stronger influence on SJA’s effective elimination of the separation bubble than the forcing amplitude did.     

Coherent structures produced by the interaction between synthetic jets and a laminar boundary layer and their surface shear stress patterns

In this paper, the results from 3D numerical simulations of circular synthetic jets issued into a zero-pressure-gradient laminar boundary layer developing along a flat plate are reported. The simulations are undertaken using FLUENT at a wide range of actuator operating conditions. The formation and development of the coherent structures produced as a result of the interaction between the synthetic jets and the boundary layer were examined using the Q-criterion. Non-dimensional parameter space maps were established to illustrate the variations in the appearance of these resultant structures and their shear stress footprints upon the changes in the operating conditions of synthetic jets. Finally, the parameter boundary separating the two distinct types of vortical structures and surface shear stress patterns is identified. It is found that the location of this boundary correlates closely with the jet-to-freestream velocity ratio of VR = 0.4 when the Strouhal number (Str) is less than 1, whereas for Str > 1 the boundary deviates from this trend, approaching the line of dimensionless stroke length of L = 1.6. In order to investigate the potential impact of the synthetic jets on the boundary layer, the increase in the space- and time-averaged skin friction coefficient relative to the baseline case without the synthetic jets is calculated. It appears that in order to maximise the impact on the near-wall flow while keeping the energy expenditure down, it is wise to maximise the accumulated effect of hairpin vortices by keeping the spacing between consecutive hairpin vortices similar to the local boundary layer thickness upstream of the separated flow instead of producing stronger individual structures.     

On the stability of almost parallel boundary layer flows

The spatial stability of a two-dimensional boundary-layer flow along a flat plate, including its non-parallel character, is determined by means of a multiple scale approximation. This leads to inhomogeneous Orr-Sommerfeld equations for the corrections due to the non-parallel effect. These equations have been solved by the method of order reduction. A neutral curves of total amplifications have been calculated based on the kinetic energy of the disturbance. The non-parallel effect decreasesthe stability.     

Turbulent modeling of a vortex boundary layer flow

The boundary layer flow of an atmospheric vortex was analyzed by utilizing a two-equation model of turbulence together with the conventional boundary layer equations. Steady-state solutions were obtained from the time-dependent solutions by using a predictor-corrector, multiple iteration method. Surprisingly complicated flow patterns with four circulating regions (4-cell vortex) were predicted in a meridional plane of the boundary layer. The intrinsic nature of sharp-turning, large-shear, and high-swirl flow characteristics may provide explanations for the tremendous roars and extremely intensive damage to ground structures associated with tornadoes. The present results are in good agreement with the tornado dust clouds simulated in our laboratory as well as those observed in natural cases.     

On an assumption guaranteeing boundary layer convergence of singularly perturbed systems

In a recent paper by B. A. Francis, an assumption was made that guaranteed convergence in the boundary layer for singularly perturbed linear systems, but the assumption was not discussed. This note examines this assumption in some detail. Verifiable sufficient and necessary conditions are given. Illustrative examples are presented.     

Thermal response of an unsteady laminar boundary layer on a flat plate due to step changes in wall temperature and in wall heat flux

The prediction of the entire time-history of the heat transfer process due either to a step change in plate surface temperature or to heat flux is studied for two-dimensional, incompressible laminar flows. The study utilizes a very efficient numerical method for solving the nonsteady energy equation. Numerical results for several Prandtl numbers are compared with analytical results obtained with the restriction of small and large time values, and the range of applicability of these analytical solutions is established.     

Laminar boundary layer on a flat plate with rotating cylinder

In the preseparation region a problem on the incompressible three-dimensional boundary layer development on a flat plate with an obstacle is posed. The uniqueness of the problem for a model equation is proved. It is shown that the outflow surface in the boundary layer on the plate is absent at the cylinder rotation. In this case there exists the special cylindrical surfaces (V-surfaces) whose rulings are orthogonal to the plate surface. The velocity vectors on each such ruling lie in the same plane, however this plane does not touch the V-surface. An absolute stable two-layer implicit difference weighted scheme of the second order approximation in all the variables is suggested which requires no fitting of the outflow surface. The comparison of the computational results with ones obtained by other authors and experiment is carried out. It is shown that besides the case of the cylinder rotation the V-surface takes place also in the case of the swept leading edge without rotation.     

Turbulent boundary layer structure of flow over a smooth-curved ramp

The effects of a convex-curved wall followed by a recovery over a flat surface on a turbulent boundary layer structure are addressed via large-eddy simulation (LES). The curved wall constitutes a smooth ramp formed by a portion of circular arc. The statistically two-dimensional upstream boundary layer flow is realistically fed by an injected inflow boundary condition. The inflow is extracted from a simultaneously simulated flat-plate boundary layer which is computed based on a compressible rescaling method. After flowing over the curved surface the flow is allowed to recover its realistic condition by passing over a downstream flat surface. The Reynolds number introduced at the inlet section of the computational domain which starts 4 times the ramp length (L_r) upstream of the curved surface is Re_δo=U_∞δ_o/ν=9907. The Reynolds number is based on the inflow boundary layer thickness δ_o, the free-stream velocity U_∞ and the kinematic viscosity ν.Mean flow predictions obtained using the present LES with the rescaling–recycling inflow condition agree well with the available experimental data from literature. The Reynolds stress components match the experimental one. However, small deviation occurs due to the smaller-domain height used in the present simulation. The experiments showed that there is a generated pressure gradient on the upper wall and this in return affects the turbulence energy on the other wall. The numerical data as well as the experiments show an enhancement of the turbulent stresses in the adverse pressure gradient region. The increased level of turbulent stresses is accompanied with large peaks aligned with the inflection point of the velocity profiles. The high stress levels are nearly unchanged by reattachment process, decaying only after the mean velocity recovered and the high production of turbulence near the outer layer drops. The recovery of the outer layer is due to the turbulent eddies generated by the separation region. Numerical visualizations show strong elongation and lifting of eddies in the region of the adverse pressure gradient generated by the curved wall. Computations of two-point correlations are also performed to represent the formation and deformation of the turbulent eddies before, over and after the curved wall. Different effects on the eddy size and its structure angle are presented.     

Analysis of the turbulent boundary layer on a rotating disk

 An order of magnitude analysis of the equations of motion governing the turbulent boundary layer on a rotating disk was performed in a stationary coordinate system located at the center of the disk. The governing equations in the wall layer were derived. Using the log law for the tangential velocity, a relation between the tangential friction velocity and the Reynolds number was derived. The forms of the three components of mean velocity in the viscous sublayer were derived. Analysis of the radial momentum equation indicated the existence of two velocity scales that are both dependent on the Reynolds number. This is the reason for the lack of inner scaling observed for the mean radial velocity. The far-wall behavior of the mean tangential and vertical velocity was documented. The analysis, along with measurements, indicated that the mean downward vertical velocity induced at the edge of the boundary layer is on the order of 15% of the tangential shear velocity. Received: 5 July 2001/Accepted: 17 October 2001     

Discrete radiation boundary conditions for a semi-infinite layer of fluid

The paper deals with the discrete formulation of radiation boundary conditions for a layer of fluid. The problem is examined with the help of the finite difference method. The proposed radiation boundary enables us to replace an infinite layer by a finite domain. The conditions ensure near equivalence between the infinite layer and the proposed finite model. The method is consistent itself and operates on a finite number of points. The results of numerical solutions are in good agreement with the results of analytical solutions of the problem.     

An asymptotic investigation into the stabilization effect of suction in the rotating-disk boundary layer flow

The suction effects in the three-dimensional boundary layer flow due to a rotating disk are analyzed from the linear stability point of view making use of the asymptotic structure of the suction mean velocity profiles of the boundary layer. The primary interest of the current work is in giving an explanation to the well-known stabilization influence of the suction from an easy to implement asymptotic means. As a consequence of the analysis, the shapes of the linear amplitude functions are derived analytically. There also results a dispersion relation for the eigenvalues existing in the limit of large suction. A comparison is then made between the perturbations obtained from the present work and also from the direct numerical solution of the linear stability equations. The asymptotic approach pursued provides a good indication as to why the large suction in the specific three-dimensional boundary layer should act in favor of the stabilization of the flow by strongly damping the external disturbances received into the suction boundary layer.