Numerical And Theoretical Investigations Of Pressure And Its Derivatives In Compressible Turbulent Flows

Investigation of pressure related objects in compressible turbulence is of great importance for the practical and fundamental interests,for example the high-speed vehicle development and turbulence modelling.In this dissertation,statistical properties of pressure fluctuations and the second order spatial derivatives of pressure(i.e.pressureHessian tensor)in compressible turbulent channel flows are studied by direct numerical simulations.With the analytical modeling of pressure-Hessian tensor,a stochastic model of velocity gradient dynamics in compressible turbulence is derived.The results and conclusions are briefly given as follows:(1)The compressibility effects on pressure fluctuation statistics in compressible turbulent channel flows are investigated.A pressure splitting method is utilized to isolate the pressure fluctuation related to compressibility,which is thus labelled as compressible pressure.It is found that the pressure and its wall-normal gradient intensities are weakened in the higher Mach number flow case and have peaks in the near wall region.The skewness and flatness of pressure gradient are also weakened obviously.These intriguing statistics behave quite different from their incompressible counterparts.It is attributed to the positive/negative staggered structures of compressible pressure which have small streamwise length scale and inhabit in the low speed streaks of viscous sublayer.The analysis based on eigenvalue spectrum shows that the positive/negative staggered structures correspond to the acoustic mode aroused in the region of locally supersonic flow in a reference frame moving at the velocity of an instability wave.Such locally supersonic region exists at flow conditions of high Mach number,low speed and low temperature.The norm distribution of pressure disturbance for optimal transient growth mode is the same as that of root mean square of compressible pressure in the near wall region of high Mach number flow case,showing that the acoustic mode dominates the compressible pressure and behaves as the agent of compressibility effect.(2)The direct numerical simulation database of a compressible turbulent channel flow with Mach number 1.56 is studied in detail,including the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor,as well as the mechanism of the pressure-Hessian tensor contributing to the evolution of invariants of the velocity gradient tensor.The results show that the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor in the central region of the channel are consistent with that of isotropic turbulence.However,in the buffer layer with relatively stronger inhomogeneity and anisotropy,the vorticity tends to be aligned with the first or second eigenvector of the pressure-Hessian tensor in the unstable focus/compressing topological region,and tends to be aligned with the first eigenvector of the pressure-Hessian tensor in the stable focus/stretching topological region.In the unstable node/saddle/saddle and stable node/saddle/saddle topological regions,the vorticity prefers to lie in the plane of the first and second eigenvectors of the pressure-Hessian tensor.The strain-rate and the pressure-Hessian tensors tend to share their second principal direction.Moreover,for the coupling between the pressure-Hessian tensor and the principal strain rates,we clarify the influence on dissipation,the nonlinear generation of dissipation and the enstrophy generation.The decomposition of the pressure-Hessian tensor further shows that the slow pressure-related term dominates the pressure-Hessian tensor’s contribution,and the influence of inhomogeneity and anisotropy mainly originates from the inhomogeneity and anisotropy of the fluctuating velocity.These statistical properties would be instructive in formulating dynamical models of the velocity gradient tensor for wall turbulence.(3)A stochastic model of velocity gradient dynamics for compressible isotropic turbulence is theoretically constructed.A set of stochastic equations of velocity gradient and conditional averaged pressure-Hessian tensor are proposed to handle the difficulty of velocity gradient dynamics modelling due to the coupling between thermodynamic variables and flow variables in compressible turbulence.Base on the assumptions of isentropic flow and Gaussian random fields for both the dilatational and solenoidal velocity,closure for the unclosed terms in the stochastic model equations is obtained.The resulting pressure-Hessian model exhibits a nature similar to the relax process,as it relaxes to the incompressible conditional averaged pressure-Hessian.By numerical simulations of the model equations,it is found that the present model can predict well both the alignment tendency between vorticity and the second eigenvector of strain-rate tensor and the probability density functions of eigenvalues of strain-rate tensor.Moreover,the joint probability density function of velocity gradient invariants and their dynamics can be simulated by the model properly.The pressure-Hessian effect on invariants dynamics are also obtained by the model.When the Mach number is high,the model leads to inaccurate evaluation of turbulence statistics,possible due to the ignored effects of shocklet structures that are present high Mach flow.