# NAVIER-STOKES BENCHMARK You are being evaluated on depth,...

Prompt

# NAVIER-STOKES BENCHMARK You are being evaluated on depth, correctness, and rigor in fluid dynamics and the mathematics of the Navier-Stokes equations. Answer every part. Show all derivations. State every assumption explicitly. If something is unknown or open, say so rather than bluffing. Do not skip steps with phrases like "it can be shown that". ## GLOBAL RULES - Use LaTeX for all equations. - Be explicit about index notation vs. vector notation, and say which you are using. - Whenever you give a numerical result, state its units or say it is dimensionless. - For every claim of a theorem, give the precise hypotheses (function spaces, dimension, boundary conditions, regularity). - Clearly mark anything you are not certain about with [UNCERTAIN]. - Finish with a "Self-audit" section listing the 3 places you are least confident in your answer. --- ## PART A: Derivation (25 points) A1. Starting from conservation of mass and conservation of linear momentum for a continuum (Cauchy momentum equation), derive the compressible Navier-Stokes equations for a Newtonian fluid. You must: - State the constitutive relation for the viscous stress tensor, including the bulk viscosity term and Stokes' hypothesis. - Explain why the stress tensor depends on the symmetric part of the velocity gradient only (frame indifference / no rotation contribution). - Write the full momentum equation in conservative and non-conservative form. A2. Derive the incompressible Navier-Stokes equations from A1. State exactly which assumptions are needed (constant density vs. divergence-free velocity are not the same thing; explain the difference and when each applies). Show that the viscous term reduces to ν∇²u and show where the term involving ∇(∇·u) goes. A3. Derive the pressure Poisson equation for incompressible flow. Then explain why pressure is not an independent thermodynamic variable in the incompressible case and what role it plays (Lagrange multiplier enforcing the constraint). State what boundary condition for pressure follows from a no-slip wall and why it is a consequence rather than a free choice. A4. Derive the vorticity transport equation in 3D. Identify the vortex-stretching term and show that it vanishes in 2D. Explain physically and mathematically why this changes the regularity picture. A5. Derive the kinetic energy balance for incompressible flow in a periodic domain and in a bounded domain with no-slip walls. Show the dissipation term explicitly in terms of ν and ∫|∇u|² dx. --- ## PART B: Nondimensionalization and scaling (10 points) B1. Nondimensionalize the incompressible equations using a characteristic length L, velocity U, and time scale L/U. Show that the Reynolds number is the only parameter. Give the dimensionless form of the pressure scaling for (i) high-Re flow and (ii) Stokes (low-Re) flow, and explain why they differ. B2. Starting from the nondimensional form, derive the Stokes equations and state what is lost. Explain the Stokes paradox for flow past a cylinder in 2D and how Oseen's correction addresses it. B3. Derive Prandtl's boundary layer equations from Navier-Stokes using a scaling argument with δ/L ~ Re^(-1/2). State the resulting equations, the boundary conditions, and what happens at separation (Goldstein singularity). Give the Blasius similarity equation for flat-plate flow, and state the numerical value of f''(0) to at least 3 significant figures. --- ## PART C: Exact solutions (15 points) For each solution below, state the geometry, verify rigorously that it satisfies both momentum and continuity, and state the boundary/initial conditions. C1. Plane Couette flow and plane Poiseuille flow. Include the pressure-driven case with a general pressure gradient and compute the mean velocity, wall shear stress, and Darcy friction factor f in terms of Re. C2. Hagen-Poiseuille flow in a circular pipe. Derive the velocity profile and the Poiseuille law relating flow rate to pressure drop. C3. The 2D Taylor-Green vortex on [0, 2π]² with periodic boundary conditions: u = cos(x) sin(y) F(t), v = -sin(x) cos(y) F(t), with a suitable pressure p. Derive F(t) and p(x, y, t). Show that this is an exact solution of the full nonlinear equations, and explain why the nonlinear term is a pure gradient here and so does not affect the decay. C4. The Lamb-Oseen vortex. Derive the azimuthal velocity profile from the vorticity diffusion equation with a delta-function initial condition. Give the circulation and the time-dependent core radius. C5. Identify one exact solution that is time-dependent AND has a nonzero nonlinear contribution (not just a gradient). If you do not know one, say so. --- ## PART D: Mathematical analysis (25 points) D1. State the Clay Millennium Problem for the 3D incompressible Navier-Stokes equations precisely: the domain (R³ and/or T³), the function spaces for initial data, the required smoothness and energy conditions, and the exact two alternatives to be proven. Do not paraphrase loosely. D2. State and prove (or give a rigorous sketch of) the a priori energy estimate. Then state the definition of a Leray-Hopf weak solution and what properties are known and unknown (existence, uniqueness, energy equality vs. inequality, regularity). D3. State precisely what is known in 2D: global existence and uniqueness of smooth solutions. Give a clear proof outline, identifying the key estimate (e.g., the enstrophy bound and why vortex stretching's absence closes it). D4. In 3D, state: - Local-in-time existence of strong solutions (Kato / Fujita-Kato), with the function space and time-of-existence scaling. - The Ladyzhenskaya-Prodi-Serrin condition: the precise exponents p, q with 2/p + 3/q ≤ 1 and what it implies. - The Beale-Kato-Majda criterion for the Euler equations and its analogue concept for Navier-Stokes. - The Caffarelli-Kohn-Nirenberg partial regularity theorem, including the dimension of the possible singular set (Hausdorff dimension / parabolic Hausdorff measure statement). D5. Explain the concept of "critical" and "supercritical" spaces for Navier-Stokes using scaling invariance u_λ(x,t) = λ u(λx, λ²t). Identify which of the following are critical, subcritical, or supercritical in 3D: L³, Ḣ^{1/2}, L^∞_t L²_x (energy), BMO^{-1}. Explain why the energy is supercritical in 3D and what that implies for the regularity problem. D6. Discuss what is known about non-uniqueness of weak solutions: summarize the convex integration approach and the result of Buckmaster and Vicol. State precisely what class of solutions it applies to (and does NOT apply to — Leray-Hopf?). Then discuss the Albritton-Brué-Colombo result and what it implies about uniqueness for Leray solutions with a forcing term. D7. Is there a known blow-up result for the 3D Navier-Stokes equations? Be careful and honest. Discuss Tao's averaged Navier-Stokes blow-up result and what it does and does not tell us about the true equations. Also discuss any computer-assisted or numerical blow-up claims you know of (e.g., for related equations such as Euler with boundary, Boussinesq, or via neural network–discovered unstable singularities), and state clearly their status (proven / computer-assisted proof / numerical evidence only). If your knowledge may be out of date, say so. --- ## PART E: Turbulence (10 points) E1. Derive the Reynolds-averaged Navier-Stokes (RANS) equations and the Reynolds stress tensor. Explain precisely why this leads to the closure problem. E2. Derive Kolmogorov's 1941 scales (η, u_η, τ_η) from dimensional analysis using only ν and the dissipation rate ε. Derive the -5/3 inertial-range spectrum E(k) = C ε^{2/3} k^{-5/3} by dimensional analysis. State the 4/5 law exactly, including what is a rigorous exact result vs. what is a scaling assumption. E3. Estimate the number of grid points required for a DNS of homogeneous isotropic turbulence at Re_λ = 1000, showing the scaling N³ ~ Re^{9/4}, and explain where the exponent comes from. State what intermittency does to the K41 predictions and name one correction model (e.g., She-Leveque) with its structure-function exponent formula. --- ## PART F: Numerical implementation (15 points) Write complete, runnable Python code (NumPy only; Matplotlib permitted for plotting) that does BOTH of the following. Do not use any CFD library. F1. A 2D incompressible Navier-Stokes solver on a periodic domain [0, 2π]² using a pseudo-spectral method in vorticity-streamfunction form, with: - 2/3-rule dealiasing - An integrating-factor RK4 (or equivalent) time integrator - Initial condition: the Taylor-Green vortex Validate it by comparing the numerical solution with the exact solution from C3. Report the L² and L^∞ error at t = 1 for ν = 0.01 and grid sizes N = 16, 32, 64, 128, and demonstrate the expected convergence behavior. Explain what the error plateau at small values tells you. F2. A 2D lid-driven cavity solver using a staggered (MAC) grid finite-difference/finite-volume projection (Chorin) method at Re = 100 and Re = 1000. Explain your discretization of the convective term, your pressure Poisson solver, and your boundary condition treatment. Report the location of the primary vortex center and compare against the well-known Ghia, Ghia & Shin (1982) benchmark values. State from memory what you recall of those values, and flag [UNCERTAIN] if you are not certain of the digits. For both: include a stability analysis (CFL and diffusion number limits for your scheme) and run time/complexity estimates. Include a unit test that checks the divergence-free condition to machine precision (spectral) or to solver tolerance (cavity). --- ## PART G: Error detection (bonus, 10 points) Each of the following statements contains at least one mistake. Identify every mistake and correct it. G1. "The incompressible Navier-Stokes equations are ∂u/∂t + (∇·u)u = -∇p + ν∇²u, with ∇·u = 0 giving a closed system of four equations in four unknowns, where pressure is determined by the equation of state." G2. "Leray proved in 1934 that 3D Navier-Stokes has unique global smooth solutions for all smooth initial data, and the Millennium Prize was awarded for this." G3. "In 2D, the vortex-stretching term ω·∇u causes enstrophy to grow without bound, which is why 2D turbulence has a direct energy cascade to small scales." G4. "The Reynolds number is the ratio of viscous to inertial forces, so high Re means viscosity dominates." G5. "Because the Taylor-Green vortex is a solution of the Euler equations, it is also an exact solution of Navier-Stokes with the same time dependence." G6. "Kolmogorov's 4/5 law is a scaling hypothesis that cannot be derived from the Navier-Stokes equations." --- ## OUTPUT FORMAT 1. Parts A through G, in order, with the headings above. 2. Each derivation: numbered steps with the justification for each step. 3. Code in separate fenced blocks, with sample output you expect (state clearly if numbers are expected values and not actual runs). 4. Final "Self-audit" section as required in the global rules.