Reynolds Transport Theorem

Reynolds Transport Theorem is a good starting point for deriving governing equations of quantities in a fluid.

Fundamental balance equation:

${\mbox{in}}-{\mbox{out}}+{\mbox{generation}}-{\mbox{consumption}}={\mbox{accumulation}}$

Reynolds Transport Theorem is a formal way of performing this balance over a fluid control volume, or fluid material volume, of arbitrary shape, moving at an arbitrary velocity.

Definitions and derivations of some important terms and identities used to construct Reynolds Transport Theorem are given here: Reynolds Transport Theorem Derivation

Definition

For any extensive fluid property $B$, there is a corresponding intensive property $b={\frac {\partial B}{\partial m}}$.

A material volume corresponding to a mass of fluid with the property $B$ can be written as $V_{B}^{M}(t)$

$B(t)=\iiint _{V_{B}^{M}}\rho b({\boldsymbol {x}},t)dV$

where $\rho$ is the fluid density.

The quantity of interest for the material volume is material derivative of the extensive property, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dB}{dt}} - that is, the rate of change of $B$ in the material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B}^{M}} (which is a volume whose boundaries move such that the flux of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} across the boundaries is 0, see Reynolds Transport Theorem Derivation page for details).

This quantity can be written as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dB}{dt} = \frac{d}{dt} \iiint_{V_{B}^{M}(t)} \rho b(\boldsymbol{x}, t) dV }

Because the integral is with time, the integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \iiint_{V_{B}^{M}(t)}} and the derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt}} do not commute.

However, if the integration were over a fixed volume that did not change in time (for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B,0}} , the integral and the derivative would commute.

The Jacobian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J = \frac{ d V_{B}^{M} }{ d V_{0,B} }} allows us to do this.

The volume integral can be converted as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \frac{ dB }{ dt } &=& \frac{d}{dt} \iiint_{V_{B}^{M}(t)} \rho b(\boldsymbol{x},t) dV \\ &=& \frac{d}{dt} \iiint_{V_{B,0}} \rho b(\boldsymbol{x},t) J dV \end{array} }

Next, the temporal derivative can be taken inside the integral (since they now commute):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \frac{d}{dt} \iiint_{V_{B,0}} \rho b(\boldsymbol{x},t) J dV &=& \iiint_{V_{B,0}} \left[ \frac{ d \rho b }{ dt } J + \rho b \frac{ dJ }{ dt } \right] dV \\ &=& \iiint_{V_{B,0}} \left[ \frac{ d \rho b }{ dt } J + \rho b \frac{ 1 }{ J } \frac{dJ}{dt} \right] J \end{array} }

Next, the Euler expansion formula can be used to simplify the second term in the integral to yield:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dB}{dt} = \iiint_{V_{B,0}} \left[ \frac{ d \rho b }{ dt } + \rho b \left( \nabla \cdot \boldsymbol{v}_{B} \right) \right] J dV }

and this can now be transformed back to the integral over the material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B}^{M}} to yield Reynolds' Transport Theorem:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dB}{dt} = \frac{d}{dt} \iiint_{V_{B}^{M}(t)} \rho b dV = \iiint_{V_{B}^{M}(t)} \left[ \frac{d \rho b}{ dt } + \rho b \left( \nabla \cdot \boldsymbol{v}_{B} \right) \right] dV }

Now, the definition of the substantial derivative, given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} = \frac{\partial}{\partial t} + (\boldsymbol{v}_{B} \cdot \nabla) }

can be used to put this into a different (more familiar) form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dB}{dt} = \iiint_{V_{B}^{M}(t)} \left[ \frac{ \partial \rho b }{\partial t} + \nabla \cdot \left( \rho b \boldsymbol{v}_{B} \right) \right] dV }

Green's Theorem can now be applied to this last divergence term, so that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{dB}{dt} = \iiint_{V_{B}^{M}(t)} \frac{ \partial \rho b }{\partial t} dV + \iint_{S_{B}^{M}(t)} \rho b \boldsymbol{v}_{B} \cdot \boldsymbol{n} dS }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_{B}^{M}} is the surface of the material volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{B}^{M}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{n}} is the vector normal to this surface, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dS} is a differential area on the surface.

References

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Lilly (1992). "A proposed modification of the Germano subgrid-scale closure method". Physics of Fluids A 4 (3): 633-636.