Conservation PDEs for CFD

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1. Introduction to Conservation Laws Conservation Law: A physical quantity $\phi$ is conserved if its rate of change within a control volume equals the net flux through its boundaries plus any sources/sinks inside. Integral Form: $$ \frac{\partial}{\partial t} \int_V \phi \, dV + \oint_S \mathbf{F} \cdot \mathbf{n} \, dS = \int_V Q \, dV $$ where $V$ is control volume, $S$ is its surface, $\mathbf{F}$ is flux vector, $\mathbf{n}$ is outward normal, $Q$ is source term. Differential (Conservative) Form: $$ \frac{\partial \phi}{\partial t} + \nabla \cdot \mathbf{F} = Q $$ This form is preferred in CFD due to its ability to correctly capture discontinuities (shocks). 2. Conservation of Mass (Continuity Equation) Physical Principle: Mass is neither created nor destroyed. Scalar Quantity: Density $\rho$ Flux Vector: $\mathbf{F} = \rho \mathbf{u}$ (convective flux) Source Term: $Q = 0$ (typically) Differential Form: $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$ where $\mathbf{u}$ is the velocity vector. Incompressible Flow (constant $\rho$): $$ \nabla \cdot \mathbf{u} = 0 $$ This implies that the velocity field is solenoidal. 3. Conservation of Momentum (Navier-Stokes Equations) Physical Principle: Newton's Second Law ($\mathbf{F} = m\mathbf{a}$) applied to a fluid particle. Vector Quantity: Momentum per unit volume $\rho \mathbf{u}$ Flux Vector: $\mathbf{F} = (\rho \mathbf{u})\mathbf{u} + p\mathbf{I} - \mathbf{\tau}$ $(\rho \mathbf{u})\mathbf{u}$: Convective momentum flux (tensor) $p\mathbf{I}$: Pressure force (isotropic stress) $\mathbf{\tau}$: Viscous stress tensor Source Term: $Q = \rho \mathbf{g}$ (body forces, e.g., gravity) Differential Form (Conservative): $$ \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u}\mathbf{u}) = -\nabla p + \nabla \cdot \mathbf{\tau} + \rho \mathbf{g} $$ where $\mathbf{u}\mathbf{u}$ is a dyadic product. Viscous Stress Tensor for Newtonian Fluid: $$ \mathbf{\tau} = \mu (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) - \frac{2}{3}\mu (\nabla \cdot \mathbf{u})\mathbf{I} $$ where $\mu$ is dynamic viscosity. For incompressible flow, $\nabla \cdot \mathbf{u} = 0$, so $\mathbf{\tau} = \mu (\nabla \mathbf{u} + (\nabla \mathbf{u})^T)$. Incompressible Navier-Stokes: $$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g} $$ (Non-conservative form, often used for incompressible, but can be derived from conservative form). 4. Conservation of Energy Physical Principle: First Law of Thermodynamics. Scalar Quantity: Total energy per unit volume $\rho E_t = \rho (e + \frac{1}{2} \mathbf{u} \cdot \mathbf{u})$ $e$: Internal energy per unit mass. $\frac{1}{2} \mathbf{u} \cdot \mathbf{u}$: Kinetic energy per unit mass. Flux Vector: $\mathbf{F} = (\rho E_t + p)\mathbf{u} - \mathbf{\tau} \cdot \mathbf{u} + \mathbf{q}$ $(\rho E_t + p)\mathbf{u}$: Convective energy flux + work done by pressure. $\mathbf{\tau} \cdot \mathbf{u}$: Work done by viscous forces. $\mathbf{q}$: Heat flux (due to conduction). Source Term: $Q = \rho (\mathbf{u} \cdot \mathbf{g}) + Q_h$ (work done by body forces + heat sources) Differential Form: $$ \frac{\partial (\rho E_t)}{\partial t} + \nabla \cdot ((\rho E_t + p)\mathbf{u} - \mathbf{\tau} \cdot \mathbf{u} + \mathbf{q}) = \rho (\mathbf{u} \cdot \mathbf{g}) + Q_h $$ Heat Flux (Fourier's Law): $\mathbf{q} = -k \nabla T$ where $k$ is thermal conductivity and $T$ is temperature. 5. Equation of State (for Compressible Flow) Relates thermodynamic properties ($p, \rho, T, e$). Ideal Gas Law: $p = \rho R T$ or $p = (\gamma - 1)\rho e$ $R$: Specific gas constant. $\gamma$: Ratio of specific heats ($c_p/c_v$). 6. Non-Dimensionalization (Important for CFD) Simplify equations and identify dominant physical effects. Characteristic scales: $L$ (length), $U$ (velocity), $\rho_0$ (density), $T_0$ (temperature), etc. Dimensionless parameters: Reynolds Number: $Re = \frac{\rho U L}{\mu}$ (ratio of inertial to viscous forces) Mach Number: $Ma = \frac{U}{a}$ (ratio of flow speed to speed of sound, $a = \sqrt{\gamma R T}$) Froude Number: $Fr = \frac{U}{\sqrt{gL}}$ (ratio of inertial to gravitational forces) Prandtl Number: $Pr = \frac{\mu c_p}{k}$ (ratio of momentum diffusivity to thermal diffusivity) 7. General Form of a Scalar Conservation Equation Advection-Diffusion Equation: $$ \frac{\partial (\rho \phi)}{\partial t} + \nabla \cdot (\rho \mathbf{u} \phi) = \nabla \cdot (\Gamma \nabla \phi) + S_\phi $$ $\phi$: Conserved scalar quantity (e.g., mass fraction, temperature) $\rho \mathbf{u} \phi$: Convective flux of $\phi$ $\Gamma$: Diffusion coefficient for $\phi$ $\Gamma \nabla \phi$: Diffusive flux of $\phi$ $S_\phi$: Source term for $\phi$