Inviscid flow

(Difference between revisions)
 Revision as of 10:03, 12 September 2005 (view source)Praveen (Talk | contribs)← Older edit Revision as of 11:41, 12 September 2005 (view source)Jola (Talk | contribs) mNewer edit → Line 6: Line 6: *Continuity equation *Continuity equation - $+ :[itex] \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}(\rho u_j)= 0 \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}(\rho u_j)= 0$ [/itex] Line 12: Line 12: *Momentum equation *Momentum equation - $+ :[itex] \frac{\partial}{\partial t}(\rho u_i) + \frac{\partial}{\partial x_j}(\rho u_i u_j + p \delta_{ij}) = 0 \frac{\partial}{\partial t}(\rho u_i) + \frac{\partial}{\partial x_j}(\rho u_i u_j + p \delta_{ij}) = 0$ [/itex] Line 18: Line 18: *Energy equation *Energy equation - $+ :[itex] \frac{\partial}{\partial t}(\rho E) + \frac{\partial}{\partial x_j}[(\rho E + p)u_j] = 0 \frac{\partial}{\partial t}(\rho E) + \frac{\partial}{\partial x_j}[(\rho E + p)u_j] = 0$ [/itex] Line 29: Line 29: * $E$ is the total energy per unit mass of fluid * $E$ is the total energy per unit mass of fluid - $+ :[itex] E = \frac{p}{\gamma - 1} + \frac{1}{2} \rho |u|^2 E = \frac{p}{\gamma - 1} + \frac{1}{2} \rho |u|^2$ [/itex] Line 37: Line 37: The above equations are closed by taking an equation of state, the simplest being the ideal gas The above equations are closed by taking an equation of state, the simplest being the ideal gas - $+ :[itex] p = \rho R T p = \rho R T$ [/itex]

Revision as of 11:41, 12 September 2005

A flow in which viscous effects can be neglected is known as inviscid flow. At high Reynolds numbers, flow past slender bodies involve thin boundary layers. Viscous effects are important only inside the boundary layer and the flow outside it is nearly inviscid. If the boundary layer is not separated then the inviscid flow model can be used to predict the pressure distribution with reasonable accuracy.

Governing Equations

The governing equations for inviscid flow, also known as Euler equations, are obtained by discarding the viscous terms from the Navier-Stokes equations

• Continuity equation
$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}(\rho u_j)= 0$
• Momentum equation
$\frac{\partial}{\partial t}(\rho u_i) + \frac{\partial}{\partial x_j}(\rho u_i u_j + p \delta_{ij}) = 0$
• Energy equation
$\frac{\partial}{\partial t}(\rho E) + \frac{\partial}{\partial x_j}[(\rho E + p)u_j] = 0$

where

• $\rho$ is the density
• $u_i$ is the fluid velocity
• $p$ is the pressure
• $E$ is the total energy per unit mass of fluid
$E = \frac{p}{\gamma - 1} + \frac{1}{2} \rho |u|^2$

The above equations are closed by taking an equation of state, the simplest being the ideal gas

$p = \rho R T$

where

• $R$ is the gas constant
• $T$ is the absolute temperature