# Inviscid flow

(Difference between revisions)
 Revision as of 10:03, 12 September 2005 (view source)Praveen (Talk | contribs)← Older edit Latest revision as of 07:46, 20 August 2013 (view source) (\gamma is forgoten, because the first term of total energy is the enthalpy) (8 intermediate revisions not shown) Line 1: Line 1: - 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. + 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. Although no practical flow is inviscid, the inviscid assumption is valid if the time scales for diffusion are much larger compared to the time scales for convection, which is measured by the [[Reynolds number]]. ==Governing Equations== ==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 + 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 *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 - $+ :[itex] - E = \frac{p}{\gamma - 1} + \frac{1}{2} \rho |u|^2 + E = \frac{\gamma p}{\gamma - 1} + \frac{1}{2} \rho |u|^2$ [/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] Line 45: Line 45: * $R$ is the gas constant * $R$ is the gas constant * $T$ is the absolute temperature * $T$ is the absolute temperature + + ==Euler equations as the limit of Navier-Stokes equations== + + Have a look at this discussion:
+ http://www.cfd-online.com/Forum/fluent.cgi?read=32788 + + ==Related pages== + *[[Euler equations in conservation form]] + *[[Rotational invariance of Euler equations]] + *[[Inviscid flux jacobians]]

## Latest revision as of 07:46, 20 August 2013

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. Although no practical flow is inviscid, the inviscid assumption is valid if the time scales for diffusion are much larger compared to the time scales for convection, which is measured by the Reynolds number.

## 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
$E = \frac{\gamma p}{\gamma - 1} + \frac{1}{2} \rho |u|^2$

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

## Euler equations as the limit of Navier-Stokes equations

Have a look at this discussion: