Difference between revisions of "See the MRF development"
(is not restricted to RANS but to incompressible fluids with constant molecular viscosity; also not restricted to a rotation about the z axis; added some structure) |
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− | + | The section describes the development for the incompressible Navier-Stokes formulation in the rotating frame. | |
− | + | == Accelerations == | |
+ | |||
+ | To start, we will look at the acceleration term for a rotating frame <math> (\vec \Omega)</math>. | ||
Notation: I: inertial, R: rotating | Notation: I: inertial, R: rotating | ||
− | + | For a general vector: | |
<math>\left [ \frac{d \vec A}{dt} \right ]_I = \left [ \frac{d \vec A}{dt} \right ]_R + \vec \Omega \times \vec A</math> | <math>\left [ \frac{d \vec A}{dt} \right ]_I = \left [ \frac{d \vec A}{dt} \right ]_R + \vec \Omega \times \vec A</math> | ||
+ | |||
+ | For the position vector: | ||
+ | |||
+ | <math>\left [ \frac{d \vec r}{dt} \right ]_I = \left [ \frac{d \vec r}{dt} \right ]_R + \vec \Omega \times \vec r</math> | ||
+ | |||
+ | <math>\vec u_I = \vec u_R + \vec \Omega \times \vec r</math> | ||
+ | |||
+ | The acceleration is expressed as: | ||
+ | |||
+ | <math>\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \vec u_I}{dt} \right ]_R + \vec \Omega \times \vec u_I</math> | ||
+ | |||
+ | <math>\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \left [ \vec u_R + \vec\Omega \times \vec r \right ] }{dt} \right ]_R + \vec \Omega \times \left [ \vec u_R + \vec \Omega \times \vec r \right ]</math> | ||
+ | |||
+ | <math>\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \vec u_R}{dt} \right ]_R + \frac{d \vec \Omega}{dt} \times \vec r + \vec \Omega \times \underbrace{ \left [ \frac{d \vec r}{dt} \right ]_R }_{\vec u_R} + \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math> | ||
+ | |||
+ | <math>\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \vec u_R}{dt} \right ]_R + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math> '''Eqn [1]''' | ||
+ | |||
+ | == Navier-Stokes equations in the inertial frame with absolute velocity == | ||
+ | |||
+ | The incompressible Navier-Stokes equations in the inertial frame with constant molecular viscosity are: | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {D \vec u_I}{D t} = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> '''Eqn [2]''' | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {\partial \vec u_I}{\partial t} + \vec u_I \cdot \nabla \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {\partial \vec u_I}{\partial t} + \nabla \cdot (\vec u_I \otimes \vec u_I) - \underbrace{( \nabla \cdot \vec u_I )}_{0} \vec u_I= - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {\partial \vec u_I}{\partial t} + \nabla \cdot (\vec u_I \otimes \vec u_I) = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> '''Eqn [3]''' | ||
+ | |||
+ | == Navier-Stokes equations in the relative frame with relative velocity == | ||
+ | |||
+ | Let's look at the left-hand side of the momentum equation of Eqn [2], by taking into account Eqn [1] for the acceleration term: | ||
+ | |||
+ | <math>\frac {D \vec u_I}{D t} = \frac{D \vec u_R}{Dt} + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math> | ||
+ | |||
+ | <math>\frac {D \vec u_I}{D t} = \frac {\partial \vec u_R}{\partial t} + \vec u_R \cdot \nabla \vec u_R + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math> | ||
+ | |||
+ | <math>\frac {D \vec u_I}{D t} = \frac {\partial \vec u_R}{\partial t} + \nabla \cdot (\vec u_R \otimes \vec u_R) + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math> '''Eqn [4]''' | ||
+ | |||
+ | since <math>\nabla \cdot \vec u_R = \nabla \cdot \vec u_I = 0</math> | ||
+ | |||
+ | <math> | ||
+ | \begin{alignat}{2} | ||
+ | \nabla \cdot \vec u_I & = \nabla \cdot \left [ \vec u_R + \vec \Omega \times \vec r \right ] = 0 \\ | ||
+ | & = \nabla \cdot \vec u_R + \underbrace{\nabla \cdot \left [ \vec \Omega \times \vec r \right ]}_{0} = 0 \\ | ||
+ | & = \nabla \cdot \vec u_R = 0 | ||
+ | \end{alignat} | ||
+ | </math> | ||
+ | |||
+ | Also, it can be noted that | ||
+ | |||
+ | <math> | ||
+ | \begin{alignat}{2} | ||
+ | \nabla \cdot \nabla (\vec u_I) & = \nabla \cdot \nabla \left [ \vec u_R + \vec \Omega \times \vec r \right ] \\ | ||
+ | & = \nabla \cdot \nabla (\vec u_R) + \nabla \cdot \underbrace{\nabla (\Omega \times \vec r)}_{0} \\ | ||
+ | & = \nabla \cdot \nabla (\vec u_R ) | ||
+ | \end{alignat} | ||
+ | </math> | ||
+ | |||
+ | Eqn [3] can be written as | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {\partial \vec u_R}{\partial t} + \frac{d \vec \Omega}{dt} \times \vec r + \nabla \cdot (\vec u_R \otimes \vec u_R) + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ | ||
+ | \nabla \cdot \vec u_R = 0 | ||
+ | \end{cases} | ||
+ | </math> '''Eqn [5]''' | ||
+ | |||
+ | Eqn [5] represents the incompressible Navier-Stokes equations in the rotating frame, in terms of rotating velocities (convection velocity and convected velocity). | ||
+ | |||
+ | == Navier-Stokes equations in the relative frame with absolute velocity == | ||
+ | |||
+ | Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame. | ||
+ | |||
+ | The term <math>\nabla \cdot (\vec u_R \otimes \vec u_R)</math> can be developed as: | ||
+ | |||
+ | <math> | ||
+ | \begin{alignat}{2} | ||
+ | \nabla \cdot (\vec u_R \otimes \vec u_R) & = \nabla \cdot ( \vec u_R \otimes \left [ \vec u_I - \vec \Omega \times \vec r \right ] ) \\ | ||
+ | & = \nabla \cdot (\vec u_R \otimes \vec u_I) - \underbrace{\nabla \cdot \vec u_R}_{0} (\vec \Omega \times \vec r) - \underbrace{\vec u_R \cdot \nabla(\vec \Omega \times \vec r)}_{\vec \Omega \times \vec u_R} \\ | ||
+ | & = \nabla \cdot (\vec u_R \otimes \vec u_I) - \vec \Omega \times \vec u_R | ||
+ | \end{alignat} | ||
+ | </math> | ||
+ | |||
+ | So, the steady term of left-hand side of Eqn [5] can be written as | ||
+ | |||
+ | <math> | ||
+ | \begin{alignat}{2} | ||
+ | \nabla \cdot (\vec u_R \otimes \vec u_R) + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r & = \nabla \cdot (\vec u_R \otimes \vec u_I) - \vec \Omega \times \vec u_R + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r \\ | ||
+ | & = \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r \\ | ||
+ | & = \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times ( \vec u_R + \vec \Omega \times \vec r ) \\ | ||
+ | & = \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I | ||
+ | \end{alignat} | ||
+ | </math> | ||
+ | |||
+ | Eqn [5] can be written in terms of the absolute velocity: | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {\partial \vec u_R}{\partial t} + \frac{d \vec \Omega}{dt} \times \vec r + \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> '''Eqn [6]''' | ||
+ | |||
+ | == Summary == | ||
+ | |||
+ | In summary, for multiple frames of reference, the incompressible Navier-Stokes equations for steady flow can be written | ||
+ | |||
+ | ::{| cellpadding="8" cellspacing="1" border="3" | ||
+ | |||
+ | |- | ||
+ | | Frame | ||
+ | | Convected velocity | ||
+ | | Steady incompressible Navier-Stokes equations | ||
+ | |- | ||
+ | | Inertial | ||
+ | | absolute velocity | ||
+ | | <math> | ||
+ | \begin{cases} | ||
+ | \nabla \cdot (\vec u_I \otimes \vec u_I) = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |- | ||
+ | | Rotating | ||
+ | | relative velocity | ||
+ | | <math> | ||
+ | \begin{cases} | ||
+ | \nabla \cdot (\vec u_R \otimes \vec u_R) + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ | ||
+ | \nabla \cdot \vec u_R = 0 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |- | ||
+ | | Rotating | ||
+ | | absolute velocity | ||
+ | | <math> | ||
+ | \begin{cases} | ||
+ | \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |- | ||
+ | |} | ||
+ | |||
+ | Back to [[Sig Turbomachinery / Developments]] |
Latest revision as of 14:54, 13 June 2009
The section describes the development for the incompressible Navier-Stokes formulation in the rotating frame.
Contents
1 Accelerations
To start, we will look at the acceleration term for a rotating frame .
Notation: I: inertial, R: rotating
For a general vector:
For the position vector:
The acceleration is expressed as:
Eqn [1]
The incompressible Navier-Stokes equations in the inertial frame with constant molecular viscosity are:
Eqn [2]
Eqn [3]
Let's look at the left-hand side of the momentum equation of Eqn [2], by taking into account Eqn [1] for the acceleration term:
Eqn [4]
since
Also, it can be noted that
Eqn [3] can be written as
Eqn [5]
Eqn [5] represents the incompressible Navier-Stokes equations in the rotating frame, in terms of rotating velocities (convection velocity and convected velocity).
Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame.
The term can be developed as:
So, the steady term of left-hand side of Eqn [5] can be written as
Eqn [5] can be written in terms of the absolute velocity:
Eqn [6]
5 Summary
In summary, for multiple frames of reference, the incompressible Navier-Stokes equations for steady flow can be written
Frame Convected velocity Steady incompressible Navier-Stokes equations Inertial absolute velocity Rotating relative velocity Rotating absolute velocity