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

Investigation of pressure related objects in compressible turbulent channel flow 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 pressure-Hessian tensor in compressible turbu-lent channel flows are studied using direct numerical simulations and a stochastic model of velocity gradient dynamics in compressible turbulence is derived analytically.The results and conclusions are briefly given as follows:(1)The compressibility effects on pressure fluctuation statistics in compressible tur-bulent 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 eigen-value 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 alignment tendencies between the local pressure-Hessian and vorticity or strain-rate tensor and the pressure-Hessian effect on the dynamics of velocity gradient invariants are investigated by a statistical analysis of compressible tur-bulent channel flow at Mach number 1.5.It is demonstrated that in the channel core region,the alignment tendencies between pressure-Hessian and voriticity or strain-rate tensor are the same as those in isotropic turbulence.In the buffer layer,vorticity of unstable focus/compressing(UFC)topology tends to align with the first or second eigenvector of pressure-Hessian tensor;vorticity of stable focus-ing/stretching(SFS)topology prefers to be parallel with the first eigenvector of pressure-Hessian tensor;and vorticity of stable node/saddle/saddle(SN/S/S)and unstable node/saddle/saddle(UN/S/S)topologies tends to be located in the plane determined by the first and second eigenvectors of pressure-Hessian tensor.The strain-rate tensor prefers to share the second eigenvector with pressure-Hessian tensor.The pressure-Hessian effects on the dynamics of dissipation,its produc-tion and enstrophy production are identified for different topologies and along the conditional mean trajectories of invariants in the eigen-coordinates of strain-rate tensor.Moreover,a splitting of pressure-Hessian related terms into three parts,i.e.,contributions from rapid,slow and the remaining pressure fluctuations,is proposed.It is revealed that the contribution from the slow pressure fluctuations dominates the pressure-Hessian effect on the dynamics of velocity gradient invari-ants even in the buffer layer region.This indicates that the inhomogeneous and anisotropic effects on pressure-Hessian contribution are originated in the nature of velocity gradient dynamics.(3)A stochastic model of velocity gradient dynamics for compressible isotropic tur-bulence 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 di-latational and solenoidal velocity,closure for the unclosed terms in the stochastic model equations is obtained.The resulting 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 func-tions 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 in-accurate evaluation of turbulence statistics,possible due to the ignored effects of shocklet structures that are present high Mach flow.