Direct numerical simulation of the turbulent boundary layer over a sharp cone with 20° cone angle (or 10° half-cone angle) is performed by using the mixed seventh- order up-wind biased finite difference scheme and sixth-order central difference scheme. The free stream Mach number is 0.7 and free stream unit Reynolds number is 250000/inch. The characteristics of transition and turbulence of the sharp cone boundary layer are compared with those of the flat plate boundary layer. Statistics of fully developed turbulent flow agree well with the experimental and theoretical data for the turbulent flat-plate boundary layer flow. The near wall streak-like structure is shown and the average space between streaks (normalized by the local wall unit) keeps approximately invariable at different streamwise locations. The turbulent energy equation in the cylindrical coordinate is given and turbulent en-ergy budget is studied. The computed results show that the effect of circumferen-tial curvature on turbulence characteristics is not obvious.
Transition prediction of the supersonic boundary layer on a cone with small angle of attack and Mach number 3.5 is investi-gated under the consideration of receptivity to slow acoustic waves, as the acoustic waves are the main environmental distur-bances in a conventional, i.e. non-quiet, wind tunnel. It is shown that the e-N method can still yield fairly satisfactory results incomparison with those obtained in wind tunnel experiments, provided that the boundary layer receptivity to slow acousticwaves is properly taken into account, including the dependence of the amplitude of disturbances on the frequency andstream-wise location. Neither the conventional e-N method nor the improved e-N method can yield correct result of transitionprediction, because the receptivity mechanisms considered there are not in accord with the real situation in the wind tunnel.
The engineering computation of turbulent flows is mainly based on turbulence modeling,however,accurate aerothermal computation of hypersonic turbulent boundary layers is still a not well-solved problem. Aerothermal computation for turbulent boundary layers on a supersonic or hypersonic blunt cone with small bluntness is done firstly by using both direct numerical simulation and BL model,and seven different cases are investigated. Then the results obtained by the two methods are compared,and the reason causing the differences is found to be the incorrect assumption in the turbulence modeling that the ratio between eddy heat conductivity and eddy viscosity is constant throughout the whole boundary layer. Based on certain theoretical arguments,a method of modifying the expression of eddy heat conductivity in the region surrounding the peak location of the turbulent kinetic energy is proposed,which is verified to be effective,at least for the seven cases investigated.
DONG Ming & ZHOU Heng Department of Mechanics,Tianjin University,Tianjin 300072,China
A new idea of using the parabolized stability equation （PSE） method to predict laminar-turbulent transition is proposed. It is tested in the prediction of the location of transition for compressible boundary layers on fiat plates, and the results are compared with those obtained by direct numerical simulations （DNS）. The agreement is satisfactory, and the reason for this is that the PSE method faithfully reproduces the mechanism leading to the breakdown process in laminar-turbulent transition, i.e., the modification of mean flow profile leads to a remarkable change in its stability characteristics.
For direct numerical simulation （DNS） of turbulent boundary layers, generation of an appropriate inflow condition needs to be considered. This paper proposes a method, with which the inflow condition for spatial-mode DNS of turbulent boundary layers on supersonic blunt cones with different Mach numbers, Reynolds numbers and wall temperature conditions can be generated. This is based only on a given instant flow field obtained by a temporal-mode DNS of a turbulent boundary layer on a flat plate. Effectiveness of the method is shown in three typical examples by comparing the results with those obtained by other methods.
Parabolized stability equations （PSE） approach is used to investigate problems of secondary instability in supersonic boundary layers. The results show that the mechanism of secondary instability does work, whether the 2-D fundamental disturbance is of the first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D wave is as large as the order 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.
This is the first part of direct numerical simulation(DNS)of double-diffusive convection in a slim rectangular enclosure with horizontal temperature and concentration gradients.We consider the case with the thermal Rayleigh number of 105,the Pradtle number of 1,the Lewis number of 2,the buoyancy ratio of composition to temperature being in the range of[0,1],and height-to-width aspect ration of 4.A new 7thorder upwind compact scheme was developed for approximation of convective terms,and a three-stage third-order Runge-Kutta method was employed for time advancement.Our DNS suggests that with the buoyancy ratio increasing form 0 to 1,the flow of transition is a complex series changing from the steady to periodic,chaotic,periodic,quasi-periodic,and finally back to periodic.There are two types of periodic flow,one is simple periodic flow with single fundamental frequency(FF),and another is complex periodic flow with multiple FFs.This process is illustrated by using time-velocity histories,Fourier frequency spectrum analysis and the phase-space trajectories.
A new method for computing laminar-turbulent transition and turbulence in compressible boundary layers is proposed. It is especially useful for computation of laminar-turbulent transition and turbulence starting from small-amplitude disturbances. The laminar stage, up to the beginning of the breakdown in laminar-turbulent transition, is computed by parabolized stability equations （PSE）. The direct numerical simulation （DNS） method is used to compute the transition process and turbulent flow, for which the inflow condition is provided by using the disturbances obtained by PSE method up to that stage. In the two test cases including a subsonic and a supersonic boundary layer, the transition locations and the turbulent flow obtained with this method agree well with those obtained by using only DNS method for the whole process. The computational cost of the proposed method is much less than using only DNS method.
Transition prediction for boundary layers has always been one of the urgent problems waiting for a solution for the development of aero-space technology,yet there is no reliable and effective method due to the complexity of the problem.The eN method has been regarded as an effective method for the transition prediction of boundary layers.However,it heavily relies on experiment or experience.And in cases with three-dimensional base flow,for instance,the boundary layer on a cone with angle of attack,the result of its application is not satisfactory.The authors have found its cause and proposed the method for its improvement,which did yield the fairly satisfactory result for a given test case,and also did not rely so much on experiment or experience.However,before people can really apply this method to practical problems,more test cases have to be studied.In this paper,more test cases for the application of the improved eN method to problems of transition prediction of supersonic and hypersonic boundary layers on cones with angle of attack will be studied.The results are compared with those obtained by experiments and/or direct numerical simulations,confirming that the improved eN method is effective and reliable.We also find that there may be more than one ZARF for each meridian plane,and which one should be chosen for the eN method has been clarified.
SU CaiHong1 & ZHOU Heng1,2 1 Department of Mechanics,Tianjin University,Tianjin 300072,China
A high-order shock-fitting finite difference scheme is studied and used to do direc-tion numerical simulation (DNS) of hypersonic unsteady flow over a blunt cone with fast acoustic waves in the free stream, and the receptivity problem in the blunt cone hypersonic boundary layers is studied. The results show that the acoustic waves are the strongest disturbance in the blunt cone hypersonic boundary layers. The wave modes of disturbance in the blunt cone boundary layers are first, second, and third modes which are generated and propagated downstream along the wall. The results also show that as the frequency decreases, the amplitudes of wave modes of disturbance increase, but there is a critical value. When frequency is over the critial value, the amplitudes decrease. Because of the discontinuity of curvature along the blunt cone body, the maximum amplitudes as a function of frequencies are not monotone.
