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Three-dimensional linear secondary instability theory is extended for compressible boundary layers on a flat plate in the presence of finite amplitude Tollmien-Schlichting (T-S) waves. The focus is on principal parametric resonance responsible for the strong growth of harmonics in a low disturbance environment.
The ideas of Benney and Bergeron (1969) and Davies (1970) on nonlinear critical layers are extended, and some new nonlinear neutral modes are computed for compressible boundary layer flow. A special case of the work is when the generalized inflexion point criterion holds. Neutral modes are found for a range of phase-speeds, dependent on the Mach number, and the properties of these are discussed. As in the linear case when the flow is relatively supersonic, multiple neutral modes exist. The behavior of the neutral amplitude in some limiting cases is also considered, and it is found that the results are significantly different from that in incompressible flow when the flow is locally supersonic.
The ideas of Benney and Bergeron (1969) and Davies (1969) on nonlinear critical layers are extended, and some new nonlinear neutral modes are computed for compressible boundary layer flow. A special case of the work is when the generalized inflexion point criterion holds. Neutral modes are found for a range of phase-speeds, dependent on the Mach number, and the properties of these are discussed. As in the linear case when the flow is relatively supersonic, multiple neutral modes exist. The behavior of the neutral amplitude in some limiting cases is also considered, and it is found that the results are significantly different from that in incompressible flow when the flow is locally supersonic.
The branch cuts in the complex frequency plane (omega) that correspond to the continuous spectrum of the two-dimensional compressible boundary layer stability equations are computed by looking for solutions that are pure oscillatory in the free-stream. In the complex omega-plane there are three compressible and one incompressible branch cuts. The compressible branch cuts are given as solutions of third order polynomials. Computations are made for a range of Reynolds and Mach numbers, and also for several streamwise wavenumbers.
The behavior of spatially unstable waves in a high Reynolds number compressible laminar boundary layer is investigated by solution of the laminar two-dimensional compressible Navier-Stokes equations (solved to fourth-order accuracy) over a flat plate with a fluctuating disturbance generated at the inflow. A significant nonlinear distortion is produced, in qualitative agreement with experimental data. It is shown that increasing compressibility can significantly stabilize the flow over a flat plate, and that the mechanism of phase cancellation is a viable mechanism for the control of growing disturbances.
Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.
A compressible linear stability theory is presented for nonparallel three-dimensional boundary-layer flows, taking into account the normal velocity component as well as the streamwise and spanwise variations of the basic flow. The method of multiple scales is used to account for the nonparallelism of the basic flow, and equations are derived for the spatial evolution of the disturbance amplitude and wavenumber. The numerical procedure for obtaining the solution of the nonparallel problem is outlined.
An investigation is conducted into the determination of the credibility of interacting boundary layers in predicting compressible subsonic flows over smooth surface imperfections. The case of smooth backward-facing steps is considered. The predicted mean flows are compared with those obtained using a Navier-Stokes solver. Moreover, the linear 2-D compressible stability characteristics of both mean flows are compared. The results show that the interacting boundary-layer formulation produces accurate mean flows that yield accurate linear stability characteristics, such as growth rates and amplification factors.
An investigation is conducted into the determination of the credibility of interacting boundary layers in predicting compressible subsonic flows over smooth surface imperfections. The case of smooth backward-facing steps is considered. The predicted mean flows are compared with those obtained using a Navier-Stokes solver. Moreover, the linear 2-D compressible stability characteristics of both mean flows are compared. The results show that the interacting boundary-layer formulation produces accurate mean flows that yield accurate linear stability characteristics, such as growth rates and amplification factors.
The Goldstein-Ruban theory has been extended within the framework of Zavol'skii et al. to study the acoustic receptivity of compressible boundary layers. We consider the receptivity produced in a region of localized, small-amplitude variation in the surface temperature and compare it with the receptivity that is induced through a similar mechanism by a variation in the suction velocity at the surface. It is found that the orientation of the acoustic wave can have a significant impact on the receptivity process, with the maximum receptivity at a given sound-pressure level being produced by upstream oriented acoustic waves. At sufficiently low Mach numbers, the variation of receptivity with the acoustic-wave orientation can be predicted analytically and is the same for both surface suction and surface heating. However, as a result of the acoustic refraction across the mean boundary layer, the above dependence can become rather complex and, also, dependent on the type of surface nonuniformity. The results also suggest that the receptivity caused by temperature nonuniformities may turn out to be more significant than that produced by the mean-flow perturbations associated with strip suction.
General solution of compressible laminar boundary layer in stagnation region of blunt bodies in axisymmetric flow
A near-wall four-equation turbulence model is developed for the calculation of high-speed compressible turbulent boundary layers. The four equations used are the k-epsilon equations and the theta(exp 2)-epsilon(sub theta) equations. These equations are used to define the turbulent diffusivities for momentum and heat fluxes, thus allowing the assumption of dynamic similarity between momentum and heat transport to be relaxed. The Favre-averaged equations of motion are solved in conjunction with the four transport equations. Calculations are compared with measurements and with another model's predictions where the assumption of the constant turbulent Prandtl number is invoked. Compressible flat plate turbulent boundary layers with both adiabatic and constant temperature wall boundary conditions are considered. Results for the range of low Mach numbers and temperature ratios investigated are essentially the same as those obtained using an identical near-wall k-epsilon model. In general, the numerical predictions are in very good agreement with measurements and there are significant improvements in the predictions of mean flow properties at high Mach numbers.
Laminar flow control and drag reduction research requires accurate boundary layer solutions as input to the three-dimensional stability analysis procedures currently under development. In support of these major programs, a fourth-order accurate finite difference scheme for solving the three-dimensional, compressible boundary layer equations has been developed and is presented in this paper. The method employs a two-point scheme in the wall normal direction and second order zigzag scheme in the cross flow direction. Accurate procedures to interface with the inviscid results are also presented. The results of applying the procedure to laminar flow on wings and fuselages are presented.
Computations based on two recently developed second-order turbulence closure models are compared with a series of boundary-layer experiments and with predictions of these experiments using an algebraic mixing length model. One of the models employs an eddy viscosity, whereas the other evaluates components of the Reynolds stress tensor. For flat plates, the computations are compared with the van Driest skin-friction transformation to assess the handling of compressibility. For boundary layers in pressure gradients, four experiments at Mach 4 and one at Mach 6.7 are used as the bases for comparison. In general, both models represent mean velocities and skin friction reasonably well, but represent the turbulence shear stress less accurately.
A solution method, fourth-order accurate in the body-normal direction and second-order accurate in the stream surface directions, to solve the compressible 3-D boundary layer equations is presented. The transformation used, the discretization details, and the solution procedure are described. Ten validation cases of varying complexity are presented and results of calculation given. The results range from subsonic flow to supersonic flow and involve 2-D or 3-D geometries. Applications to laminar flow past wing and fuselage-type bodies are discussed. An interface procedure is used to solve the surface Euler equations with the inviscid flow pressure field as the input to assure accurate boundary conditions at the boundary layer edge. Complete details of the computer program used and information necessary to run each of the test cases are given in the Appendix.
A three-dimensional boundary-layer code was developed for particular application to realistic hypersonic aircraft. It is very general and can be applied to a wide variety of boundary-layer flows. Laminar, transitional, and fully turbulent flows of compressible, reacting gases are efficiently calculated by use of the code. A body-oriented orthogonal coordinate system is used for the calculation and the user has complete freedom in specifying the coordinate system within the restrictions that one coordinate must be normal to the surface and the three coordinates must be mutually orthogonal.
Turbulent shear-stress, eddy-viscosity, mixing-length, heat-flux and Prandtl number distributions across a hypersonic turbulent boundary layer have been determined from the 'time-averaged' conservation equations using experimental mean profile data obtained at several streamwise locations in a fully developed turbulent boundary layer with negligible pressure gradient. The eddy-viscosity, mixing-length and Prandtl number results show general agreement with previous incompressible and adiabatic compressible correlations. However, when the turbulent Prandtl number is defined using total enthalpy as opposed to static enthalpy no clear correlation of the results in the outer portion of the boundary layer could be obtained.
Two fourth order accurate Osher schemes are presented which maintain higher order accuracy on nonuniform grids. They use either a conservative finite difference or finite volume discretization. Both methods are successfully used for direct numerical simulations of flat plate boundary layer instability at different Mach numbers. Results of growth rates of Tollmien-Schlichting waves compare well with direct simulations of incompressible flow and for compressible flow with results obtained by solving the parabolic stability equations.