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.     

Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first order ordinary differential equations. The streamwise shear stress function θ and the cross-flow velocity component V are represented as suitable functions of the streamwise velocity component U. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30° incidence. The results are compared with corresponding finite difference solutions.     

Numerical modelling and passive flow control using porous media

The whole flow over a solid body covered by a porous layer is presented. The three main models used in the literature to compute efficiently the fluid flow are given: the reduction of the porous layer to a boundary condition, the coupling of Darcy equation with Navier-Stokes equations and the Brinkman-Navier-Stokes equations or the penalisation method. Numerical simulations on Cartesian grids using the latest model give easily accurate solutions of the flow around solid bodies with or without porous layers. Adding appropriate porous devices to the solid bodies, an efficient passive control of the two-dimensional incompressible flow is achieved. A strong regularisation of the flow is observed and a significant reduction of the vortex induced vibrations or the drag coefficient is obtained.     

A computerized analysis of supersonic nonuniform flows over sharp and spherically blunted cones at angle of attack

A system of computer programs has been developed to predict supersonic inviscid and viscous nonuniform flow fields over sharp and spherically blunted cones at angle of attack. For blunt cones the flow fields considered were axisymmetric wake flows positioned such that the flow in the subsonic nose region remained axisymmetric. For sharp cones, both axisymmetric wake flows and two-dimensional shear flows were considered. The programs used in solving inviscid flow fields incorporate a modified inverse method for solving subsonic flow regions and modified axiymmetric and three-dimensional method of characteristics procedures for solving the supersonic flow regions. Body properties predicted by the inviscid solutions were used as edge data for solution of the corresponding laminar boundary-layers over the bodies. The viscous flow solutions were obtained using axisymmetric and full three-dimensional boundary-layer programs. Typical results from inviscid calculations have shown the development of strong adverse pressure gradients over both sharp and blunt cones in wake flows. In addition a thin entropy layer was found near the surface of both bodies; however, the normal pressure gradient was found to be negligible for the nonuniform flows considered. For the sharp cone in shear flow, property variation along the body was found to be almost linear. In all cases the aerodynamic coefficients were found to be significantly affected by the free-stream nonuniformity. Typical viscous flow field results have shown that relative to uniform flow values the skin friction and heat transfer increase along the windward streamline of both blunt and sharp cones in the nonuniform flows considered. Decreasing the width of the wake in wake flow increases the heat transfer and skin friction.     

Burnett hypersonic thin shock layer near the windward side of a flat plate

A one-layer model of a Burnett thin shock layer is simplified to fit the problem of a rarefied gas hypersonic flow along the windward side of a flat plate placed at a large angle of attack with an incident flow; here, the mathematical apparatus of the theory of a two-layer hypersonic thin viscous shock layer near nonthin bodies is used. This theory suggests that in the structure of a flow, between the followed body’s surface and the incident undisturbed flow, we have two typical regions-sublayers (a smeared pressure shock plus the shock layer proper). It is shown that the considered Burnett thin shock layer problem in the two-layer approximation completely reduces to the corresponding Navier-Stokes problem.     

Bifurcations and buffet of transonic flow past flattened surfaces

Turbulent transonic flow past flattened aerodynamic surfaces is investigated numerically using the RANS equations. The study is focused on: (★) a buffet onset caused by instability of the shock wave/boundary layer interaction, (★★) instability of the entire flow structure and related flow bifurcations.For a symmetric airfoil at zero angle of attack, computations reveal both bifurcations and buffet in a range of the freestream Mach number M_∞. At nonzero angles of attack, α=±1°, there are two ranges of M_∞ in which the buffet onset takes place. For a Whitcomb type airfoil, computations demonstrate instability of the flow structure only at negative α. Axisymmetric flow past axisymmetric bodies is also considered, and instability of the flow structure at certain freestream Mach numbers is shown.     

A computational study of the effect of angles of attack on a double-cone type supersonic inlet with a bleeding system

The flow characteristics of a supersonic inlet with a bleeding system under varying angles of attack are studied by computational 3D turbulent flow analysis. A compressible upwind flux difference-splitting Navier–Stokes method with the k–ω turbulence model is used to analyze the super inlet flowfields. The bleeding system successfully removes the low energy flow from the boundary layer near the throat. The bleeding system enables the supply of more uniform flow at the engine face compared to a supersonic inlet without a bleeding system at the expense of mass flow. More non-uniform flowfields are seen at the Aerodynamic Interface Plane when the angle of attack increases. These non-uniform flowfields at the supersonic engine face degrade engine performance.     

Direct numerical simulation of flow separation around a NACA 0012 airfoil

Direct numerical simulation (DNS) for the flow separation and transition around a NACA 0012 airfoil with an attack angle of 4° and Reynolds number of 10⁵ based on free-stream velocity and chord length is presented. The details of the flow separation, detached shear layer, vortex shedding, breakdown to turbulence, and re-attachment of the boundary layer are captured in the simulation. Though no external disturbances are introduced, the self-excited vortex shedding and self-sustained turbulent flow may be related to the backward effect of the disturbed flow on the separation region. The vortex shedding from the separated free shear layer is attributed to the Kelvin-Helmholtz instability.     

Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction

The boundary layer flow of a micro-polar fluid due to a linearly stretching sheet is investigated. The influence of various flow parameters like ‘suction and injection velocity through the porous surface’, ‘viscosity parameter causing the coupling of the micro-rotation field and the velocity field’ and ‘vortex viscosity parameter’ on ‘shear stress at the surface’, ‘fluid velocity’ and ‘micro-rotation’ are studied. The governing equations of the transformed boundary layer are solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. Comparison between the HAM and numerical solutions showed excellent agreement.     

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.     

Non-body-fitted Cartesian-mesh simulation of highly turbulent flows using multi-relaxation-time lattice Boltzmann method

This paper presents a lattice Boltzmann method (LBM) based study aimed at numerical simulation of highly turbulent and largely inclined flow around obstacles of curved geometry using non-body-fitted Cartesian meshes. The approach features (1) combining the interpolated bounce-back scheme with the LBM of multi-relaxation-time (MRT) type to enable the use of simple Cartesian mesh for the flow cases even with complex geometries; and (2) incorporating the Spalart–Allmaras (SA) turbulence model into LBM in order to represent the turbulent flow effect. The numerical experiments are performed corresponding to flows around an NACA0012 airfoil at Re=5×10⁵ and around a flat plate at Re=2×10⁴, respectively. The agreement between all simulation results obtained from this study and the data provided by other literature demonstrates the reliability of the enhanced LBM proposed in this paper for simulating, simply on Cartesian meshes, complex flows that may involve bodies of curved boundary, high Reynolds number, and large angle of attack.     

Three-dimensional effects in multi-element high lift computations

In an effort to discover the causes for disagreement between previous two-dimensional (2-D) computations and nominally 2-D experiment for flow over the three-element McDonnell Douglas 30P-30N airfoil configuration at high lift, a combined experimental/CFD investigation is described. The experiment explores several different side-wall boundary layer control venting patterns, documents venting mass flow rates, and looks at corner surface flow patterns. The experimental angle of attack at maximum lift is found to be sensitive to the side-wall venting pattern: a particular pattern increases the angle of attack at maximum lift by at least 2°. A significant amount of spanwise pressure variation is present at angles of attack near maximum lift. A CFD study using three-dimensional (3-D) structured-grid computations, which includes the modeling of side-wall venting, is employed to investigate 3-D effects on the flow. Side-wall suction strength is found to affect the angle at which maximum lift is predicted. Maximum lift in the CFD is shown to be limited by the growth of an off-body corner flow vortex and consequent increase in spanwise pressure variation and decrease in circulation. The 3-D computations with and without wall venting predict similar trends to experiment at low angles of attack, but either stall too early or else overpredict lift levels near maximum lift by as much as 5%. Unstructured-grid computations demonstrate that mounting brackets lower the lift levels near maximum lift conditions.     

The direct matrix imbedding technique for computing three-dimensional potential flow about arbitrarily shaped bodies

The direct matrix imbedding technique is used to solve Laplace's equation for the velocity potential numerically about arbitrarily shaped bodies with normal gradient boundary conditions in two and three dimensions. The bodies are imbedded in Cartesian grids overlaying relatively large rectangular and box regions. Solutions are obtained only in those parts of the grid necessary for constructing solutions to potential flow problems. An important subclass of these problems, considered in this paper, is ship wave problems in channels. Uniform and stretched Cartesian grids are considered, and solutions are obtained very quickly. Results are presented.     

On strong slot injection into a subsonic laminar boundary layer

This paper is concerned with the problem of fluid injection into a steady, subsonic, laminar, boundary layer flow over a flat plate at zero angle of attack. The problem is formulated within the context of the triple deck theory for asymptotic analysis of strong slot injection for large Reynolds number. The lower deck, linearized problem for small injection velocity is solved analytically using Fourier transform techniques. Graphical results are given for a wide range of slot lengths. A numerical technique for the lower deck non-linear problem is presented together with results for various injection velocities. In contrast to the supersonic case, separation is found to first occur downstream of the slot, where a recirculating flow bubble is formed.     

Flutter prediction by an Euler method on non-moving Cartesian grids with gridless boundary conditions

A method is presented for the prediction of transonic flutter by the Euler equations on a stationary Cartesian mesh. Local grid refinement is established through a series of embedded meshes, and a gridless method is implemented for the treatment of surface boundary conditions. For steady flows, the gridless method applies surface boundary conditions using a weighted average of the flow properties within a cloud of nodes in the vicinity of the surface. The weighting is established with shape functions derived using a least-squares fitting of the surrounding nodal cloud. For unsteady calculations, a perturbation of the shape functions is incorporated to account for a fluctuating surface normal direction. The nature of the method provides for efficient and accurate solution of transient flow problems in which surface deflections are small (i.e. flutter calculations) without the need for a deforming mesh. Although small deviations in angle of attack are considered, the mean angle of attack can be large. Results indicate good agreement with available experimental data for unsteady flow, and with computational results addressing flutter of the Isogai wing model obtained using traditional moving mesh algorithms.     

Laminar flow in rectangular channels. Part I: Entry analysis. Part II: Numerical solution for a square channel?

Laminar incompressible flow in rectangular channels is considered. In Part I, the entry region is evaluated by a boundary layer/potential core analysis. It is shown that the three-dimensional displacement induced potential flow can be described with a pair of two-dimensional potential functions. Second-order boundary layer solutions, with and without surface mass transfer, are determined; an interesting secondary flow reversal is predicted. In Part II, numerical solutions are obtained for the viscous channel equations, which are derived from the asymptotic theory of Part I. A two stream function, velocity, vorticity system, independent of the Reynolds Number, is solved with a combined iterative ADI/point-relaxation numerical procedure. A single calculation applied for all Reynolds numbers, which appears only in the coordinate scaling. The axial flow behavior of Parts I and II are in good agreement in the asymptotic entry region where both analysis apply. Secondary flow reversal is calculated; however, the grid is too crude for quantitative comparisons. Numerical solutions are obtained until fully developed conditions are achieved. Agreement with experimental data is good.     

An implicit method for three dimensional viscous flow with application to cones at angle of attack

An iteration method for solving the implicit difference equations associated with three nonlinear parabolic differential equations is derived and analyzed. The method is applied to the high Reynolds number viscous flow around a cone at high angle of attack. The requirements which must be met to ensure convergence of the iterations are obtained. In addition, an analysis of the stability of the difference equations is presented and discussed. The numerical results are compared with experimental data for a 10° cone at 12° angle of attack, and a 5·6° cone at 8° angle of attack. The agreement is very good.     

Drag minimization on a body of revolution through evolution

This study deals with the determination of the minimum drag profile of a body of revolution. With the restriction that the boundary layer is not allowed to separate, the profile is described by a continuous, linear line-singularity distribution defined at a number of points placed parallel in a uniform stream with zero angle of attack. The drag calculation is based on the momentum deficit in the boundary layer at the trailing edge of the body. By gradually changing the body profile with the Evolution Strategy the minimum drag profile is found for a constant fineness ratio. For a particular case, the drag coefficient is reduced by 30 percent by means of this procedure.     

分布抽吸率对整车风洞试验段流场影响的数值模拟

为探讨适用于整车风洞的分布抽吸率,采用计算流体力学（Computational Fluid Dynamics,CFD）方法研究经边界层控制后的试验段流场．针对不同分布抽吸率下的试验段,利用FLUENT对流场进行模拟,然后计算边界层位移厚度、静压因数和气流偏角．二维和三维的数值模拟结果均表明,试验段边界层的位移厚度随分布抽吸率的增加而减小;当分布抽吸率达到某一数值后,若继续增大,那么边界层的位移厚度将基本保持不变．研究结果对风洞建设及试验时分布抽吸率的选取有参考意义．     

The flow induced in a three layer stratified fluid by a submerged sink or source with stagnation points on the free surfaces

A boundary integral technique is developed to study the free surface flow of a steady, two-dimensional, incompressible, irrotational and inviscid fluid which is induced in both two and three layer stratified fluids in the presence of gravity by a submerged sink or source with stagnation points on the free surfaces. A special form of the Riemann–Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing non-linear boundary integral–differential equations which have been solved for the fluid velocity on the free surfaces and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various Froude numbers and sink heights in both two and three layer fluids. Further, we have also studied the critical Froude numbers for which no convergent solutions are possible for any larger values of the Froude number. We have found that the free surfaces are dependent on two parameters, namely the Froude number and the ratio of sink height to the thickness of either the middle layer in a three layer system and the bottom layer in a two layer system.