- 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