Questions and Answers of Navier Stokes Turbulen

Derive the dimensional vorticity pde (2.62) for a single, compressible Newtonian fluid using Cartesian coordinates.pde (2.62) 1 D Wa = wsa + 5 EaY 1 + Eay + Eay 1 Fr ax Dt
Verify that (2.69) is the solution of the vorticity pde (2.65) in \(\mathcal{D}=R^{3}\) for inviscid and barotropic fluids.pde (2.69)pde (2.65) (T, X) R(T, X) (0, X) a (T, X) R(0, X) OXB
Consider the flow of an inviscid, incompressible fluid with non-zero vorticity governed by the Euler pdes in \(\mathcal{D}=R^{3}\), let the vorticity be specified initially as a smooth vector field
Consider the flow of an incompressible, Newtonian fluid. Define the Lamb vector by Eq. (2.54) \(\mathbf{L} \equiv \omega \times \mathbf{v}\), where \(\boldsymbol{\omega} \equiv abla \times
Determine the symmetries of the heat pde\[ \frac{\partial T}{\partial t}-\frac{\partial^{2} T}{\partial x^{2}}=0 \]defined on \(\mathcal{D}=[0, \infty] \times R^{1}\).2.5.1 Change the notation to
Integrate the Euler pdes (2.9)\[ \begin{gathered} \frac{\partial v_{\alpha}}{\partial x_{\alpha}}=0 \\ \frac{\partial v_{\alpha}}{\partial t}+v_{\beta} \frac{\partial v_{\alpha}}{\partial
Derive the pde for the dimensionless mean kinetic energy \(k \equiv \frac{1}{2}\left\langle v_{\alpha}^{\prime} v_{\alpha}^{\prime}\rightangle\) assuming a viscous, incompressible Newtonian fluid.
Derive the pde for the dimensionless mean enstrophy \(\left\langle e^{2}\rightangle\) defined by (3.16) in homogeneous turbulence for an incompressible Newtonian fluid with constant viscosity
Consider the pde for the mean enstrophy obtained in the previous problem.(3.3.1) Solve the pde assuming that the vortex stretching term has the form\[ \left\langle\omega_{\alpha} \omega_{\beta}
The Jacobi orthogonal polynomials \(P_{n}^{\alpha, \beta}(x)\), which are the solutions of the ode\[ \left(1-x^{2}\right) \frac{d^{2} P_{n}^{\alpha, \beta}}{d x^{2}}+[\beta-\alpha-(\alpha+\beta+2)
The Hermite functions (Sect. 4.3.3) are a basis for the space \(\Omega=\) \(C_{R^{1}}^{\infty} \cap L_{R^{1}}^{2}\) of functions defined on the unbounded domain \(R^{1}\). Determine the coordinates
Solve the BVPs for the Poisson pdes\[ \epsilon_{\delta \eta \alpha} \frac{\partial \omega_{\alpha}}{\partial x_{\eta}}=\frac{\partial^{2} v_{\eta}}{\partial x_{\delta} \partial
Let \(\Omega=\left\{\mathbf{v}(\mathbf{x}) \in L_{\mathcal{D}}^{2} \cap C_{\mathcal{D}}^{2}\right\}\) be the phase space for the compact flow domain \(\mathcal{D} \subset R^{3}\) with n.e. smooth
Compute the Wiener integral of the functional\[ \begin{equation*} F[f(x)]=\exp \left\{\lambda \int_{0}^{1} d y w(y) f^{2}(y)\right\} \tag{6.42} \end{equation*} \]directly (without using the
Compute the Wiener integral (6.39) for \(p(x)=1, q(x)=0\) and \(w(x)=(x+\alpha)^{-2}\), where \(0Eq (6.39) [ exp(x dxw(x) f(x) ditu (f) = (()) C 0 dw
Consider the space of continuous functions \(C=\{f(x), x \in[0,1]\}\) defined on the domain \(\mathcal{D}=[0,1]\). Solve the IVP for the functional differential equation\[ \frac{\delta F[f]}{\delta
Consider the space of twice continuously differentiable functions \(C^{2}=\{f(x), x \in[0,1]\}\). Solve the functional differential equation\[\frac{\delta F[f]}{\delta f(x)}=\left[-\frac{d^{2} f}{d
Solve the pure IVP for the Burgers pde (1.2) with initial condition \(u(0, x)=u_{0}(x) \in L_{\mathcal{D}}^{2} \cap C_{\mathcal{D}}^{\infty}, \mathcal{D}=(-\infty, \infty)\), using the Hopf-Cole
Derive the Hopf fde for the characteristic functional \(\theta[y ; t]\) for the pure IVP of the Burgers pde (1.2). Use the result obtained in Problem (9.1) to establish the solution operator and its
Solve the Hopf fde for the Burgers pde using the solution operator established in Problem (9.2). The initial condition is the characteristic functional \(\theta[y ; 0]\) for Gaussian stochastic
Consider an analytic functional \(R[y]\) defined on the phase space \(\Omega=\) \(\left\{y(\mathbf{x}) \in L_{R^{3}}^{2}\right\}, \mathbf{x} \in R^{3}\) and Gaussian stochastic fields \(f_{i}(t,
Solve the fde for the characteristic functional \(\theta[\mathbf{y}() ; t\).\[ \frac{\partial}{\partial t} \theta[\mathbf{y} ; t]=i \int_{\mathcal{D}} d \mathbf{x}
Transform the Hopf fde (9.40) to cylindrical coordinates in \(\mathcal{D}\). The explicit form of the pressure gradient term as functional of the velocity is not required. 1 82 80 [y; t]+illaly, g;
Apply the conditions of steady state and solenoidal vector fields \(\mathbf{y}\) to the Hopf fde for cylindrical coordinates in the flow domain \(\mathcal{D}\) obtained in the previous Problem
Consider the flow through a straight, circular pipe periodic in axial direction. A solenoidal ONS basis \(\mathcal{B}_{e}\) was constructed in Sect. 25.21 spanning the test function space
A random variable \(Y(t)\) defined on \(R^{1}\) is specified by the Pdf\[ f_{Y}(t)=\frac{1}{2} \exp (-|t|) \]Compute the characteristic function \(\theta(\zeta)\) and all statistical moments using
Consider the random variable \(\epsilon>0\) such that\[ \Phi=\ln \left(\frac{\epsilon}{\epsilon_{0}}\right) \]is Gaussian with mean \(\langle\Phiangle\) and variance \(0
Consider the dynamics of a passive scalar \(0 \leq \Phi(t, \mathbf{x}) \leq 1\) in a turbulent flow. Specialize the transport pde (11.52) for the \(\operatorname{Pdf} f_{N}\left(\varphi_{1}, \ldots,
Compute the Gateaux derivative with respect to \(v_{\alpha}(\mathbf{x}, t)\) of the pressure represented in terms of the Green's function as shown in Sect. 11.1 for nonhomogeneous Neumann conditions
The initial value problem (IVP) for the ode\[ \frac{d Y}{d t}=-C Y^{2} \]with \(C\) being a positive constant and initial condition \(Y(0)=Y_{0} \geq 0\) generates a mapping \(Y(t): Y(0)
Derive the single label Pdf equation for position \(\Phi_{\alpha}(\tau, \mathbf{X})\) in the material description. Use the coarse-grained Pdf\[ \hat{f} \equiv \prod_{\alpha=1}^{3}
Compute the analytic solution of the IVP for the mapping pde (13.23) for a single conserved \((Q(\eta)=0)\) scalar. The scalar space is the unit interval \(\mathcal{R}_{\Phi}=\) \([0,1]\), and the
Compute and plot the Jacobian \(J(\eta ; t)\) and the \(\operatorname{Pdf} f(\varphi ; t)\) for \(t>0\) using the mapping \(\mathcal{X}(\eta ; t)\) of the previous problem 13.1.Problem 13.1Compute
Consider the discriminating scalar \(\Phi(t, \mathbf{x}) \geq 0\) governed by\[ \frac{\partial \Phi}{\partial t}+v_{\alpha} \frac{\partial \Phi}{\partial x_{\alpha}}=\frac{\partial}{\partial
Compute the intermittency generating vector (16.19)\[ \varphi_{\alpha}(\mathbf{w} ; \tau, \Delta \tau, \mathbf{X})=\frac{1}{\sqrt{\Sigma_{(\alpha \alpha)}}}\left\langle\Delta A_{\alpha} \mid
Consider stationary and locally isotropic turbulence at high Renumber. A simple model for the energy spectrum can be constructed by defining a sequence of wavenumbers \(k_{n}\), such that\[ k_{n+1}=2
Determine the pde for helicity density \(h(t, \mathbf{x}) \equiv \mathbf{v} \cdot !\) for the flow of an incompressible Newtonian fluid. Define helicity \(\mathcal{H}\) by (15.30) and establish the
Compute the solution \(\Psi\) of the pde (20.26) for the Chaplygin-Lamb dipole streamfunction\[ \frac{\partial^{2} \Psi}{\partial r^{2}}+\frac{1}{r} \frac{\partial \Psi}{\partial r}+\frac{1}{r^{2}}
Compute the solution \(\Psi\) of the pde (20.26) for the Chaplygin-Lamb dipole streamfunction\[ \frac{\partial^{2} \Psi}{\partial r^{2}}+\frac{1}{r} \frac{\partial \Psi}{\partial r}+\frac{1}{r^{2}}
Determine the material deformation gradient as measured by deformation gradient and deformation rate (velocity gradient) for the restricted Euler system. The restricted Euler flow is governed by
One of many Lagrangean line structures in turbulent flows is considered in elementary form. The example, called Shilnikov system, is constructed as simplified velocity field that contains several
Townsend's model eddy (19.1) is a localized blob of vorticity defined in \(\mathcal{D}=R^{3}\) (Davidson [76], Sect. 6.4.1, cylindrical coordinates) by \(v_{r}=v_{z}=0\) and\[ v_{\theta}=\Omega r
Consider the fully developed, turbulent flow of an incompressible, Newtonian fluid through a plane channel between two plates at \(x_{2}=0\) and \(x_{2}=h\). Use dimensionless variables based on the
Consider a materially invariant admissible circuit \(\mathcal{C}\), as defined in Sect. 2.6.3, embedded in a flow field \(\mathcal{D}\). The incompressible fluid is in turbulent motion governed by
Orlandi and Carnevale [37] argue that the nonlinear amplification of vorticity in inviscid interaction is a candidate for the appearance of a finite-time singularity of the second kind starting from
Show that the Duchon-Robert smoothness term \(D(\mathbf{v})\) in Sect. 22.3 is zero for the following class of velocity fields \(\mathbf{v}(\mathbf{x}, t)\) :\[ \int_{\mathcal{D}} d