Probability distribution function and parallel numerical simulation in turbulence

1. We have theoretically and numerically studied可能性density function (PDF) Q(ξ) for thevelocity gradient ξ =Ξ θu/θx in the forced one dimensional Burgers equation which is well known as a简单模型for the Navier-Stokes turbulence as well as a model for the growth of randominterface. It is found that Q(ξ) is very skewed and has an algebraic tail followed by theexponential decrease for large negative ξ.2.我们有表现非常大的比例DNSs of the 3dincompressible turbulence using high performance vector parallel machine (Fujitsu VPP5000) with 32教授在Nagoya University. A very efficient FFT for the parallel machine has been developedfor this purpose. With the resolution N = 1024we have attained the Raynolds number R D2λ D2 = 490，the highest value and world record at present，and obtained various notable findings. The results have been talked at The internationalconference on turbulence at ITP， UCSBattracted very strong interests of the audience(http://online.itp.ucsb.edu/online/hydrot_c00).3. physics or conditional averages of the dissipationterm of the Navier-Stokes equation for a given value of δu(r) = u(x+r)-u(x) gives us some insightsAsymptotic forms of Q(δu) in 3d steady homogenous turbulence are found to beGaussian for small |δu| except prefactor and exponential for large |δu|，4.我们曾使用过各种不同的pressure和itsgradient in the 3d homogenous turbulence by the DNS. It is found that their values and Reynoldsclassical Kolmogorov theory: the number dependence are different from the ones predicteddifference can be explained in the coherent structure of the source term in the Poissonequation for the pressure. Also found is a new scaling for the pressure spectrum. One of theimportant findings是一个普遍的概念of the pressure spectrum in the Kolmogorov scaling as4.48.

1. We have theoretically and numerically studied probability density function (PDF) Q(ξ) for the velocity gradient ξ =Ξ θu/θx in the forced one dimensional Burgers equation which is well known as a simple model for the Navier-Stokes turbulence as well as a model for the growth of random interface. It is found that Q(ξ) is very skewed and has an algebraic tail followed by the exponential decrease for large negative ξ.2. We have performed very large scale DNSs of the 3D incompressible turbulence using high performance vector parallel machine (Fujitsu VPP5000) with 32 processors at Nagoya University. A very efficient FFT for the parallel machine has been developed for this purpose. With the resolution N = 1024ィイD13ィエD1, we have attained the Raynolds number RィイD2λィエD2 = 490, the highest value and world record at present, and obtained various notable findings. The results have soon been talked at the international conference on turbulence at ITP, UCSB, and attracted very strong interests of the audience (http://online.itp.ucsb.edu/online/hydrot_c00).3. Physics or conditional averages of the dissipation term of the Navier-Stokes equation for a given value of δu(r) = u(x+r)-u(x) gives us some insights into the PDf Q(δu). Asymptotic forms of Q(δu) in 3D steady homogenous turbulence are found to be Gaussian for small |δu| except prefactor and exponential for large |δu|, which is in good agreement with DNS.4. We have examined the variances of the pressure and its gradient in the 3D homogenous turbulence by the DNS. It is found that their values and Reynolds number dependence are different from the ones predicted by classical Kolmogorov theory. The difference can be explained in terms of the coherent structure of the source term in the Poisson equation for the pressure. Also found is a new scaling for the pressure spectrum. One of the important findings is a universal constant of the pressure spectrum in the Kolmogorov scaling as 4.48.