Problem Statement: Starting from the basic conservation laws, derive the incompressible Navier-Stokes equations in vector form. Explicitly state the physical meaning of each term in the final equation.
Solution: We begin with the conservation of mass (Continuity Equation) and conservation of momentum.
Problem:
For a power-law fluid: ( \tau_rz = K \left| \fracdudr \right|^n-1 \fracdudr ) (( n>0 )), laminar steady flow in a circular pipe of radius ( R ) driven by pressure gradient ( -\fracdpdz = G > 0 ). Find the velocity profile and total flow rate. advanced fluid mechanics problems and solutions
Scenario: Airflow over an airfoil near stall. The pressure increases downstream (adverse gradient), threatening flow separation.
Key Equations: Time-averaged Navier-Stokes (RANS) introduces the Reynolds stress tensor (\rho \overlineu_i' u_j'). Problem: For a power-law fluid: ( \tau_rz =
Challenge: Closure problem—we have more unknowns than equations.
Solution Strategies:
Example Solution: For a NACA 4412 airfoil at ( \alpha = 12^\circ ), use LES with a dynamic Smagorinsky subgrid-scale model. Validate against experimental (C_p) (pressure coefficient) distributions. The solution reveals a laminar separation bubble followed by turbulent reattachment—a phenomenon impossible to capture with RANS alone.
The Problem: A square cavity with top lid moving at velocity ( U ), other walls stationary. Solve for the stream function and vorticity distribution. Example Solution: For a NACA 4412 airfoil at
The Numerical Solution Framework:
The solution reveals that the primary vortex moves toward the geometric center as Re increases, and tertiary vortices appear at Re > 5000.