Hydrodynamics Entropy Function

from Trangenstein, 4.4.2 through 4.4.4

Thermodynamics

ρ

= density

P

= pressure

e_{i n t}

= internal specific energy

T

= temperature

S

= entropy

i, j, k, . . .

spans

x, y, z

I, J, K, . . .

spans all elements

Second law of thermodynamics:

d e_{i n t} = T d S - P d (1 / ρ)

therefore

P = \partial_{1 / ρ} e

T = \partial_{S} e_{i n t}

Ideal gas relation:

P = ρ R T

e_{i n t} = c_{v} T

where

R

= gas constant

c_{v}

= heat capacity
polytropic gas heat capacity:

c_{v} = \frac{R}{γ - 1}

where

γ

= heat capacity ratio

polytropic ideal gas:

e_{i n t} = c_{v} T = \frac{R T}{γ - 1}

therefore

(γ - 1) e_{i n t} = R T

therefore

P / ρ = R T = (γ - 1) e_{i n t}

P = (γ - 1) ρ e_{i n t}

specific enthalpy:

h = e_{i n t} + P / ρ = e_{i n t} + (γ - 1) e_{i n t} = γ e

where

h

= enthalpy

second law, in terms of

d S

d S = \frac{1}{T} (d e_{i n t} + P d (1 / ρ))

d (1 / ρ) = - 1 / ρ^{2} d ρ

therefore

d S = \frac{1}{T} d e_{i n t} - \frac{1}{T ρ^{2}} P d ρ

second law for polytropic ideal gas:
using

e_{i n t} = \frac{P}{ρ (γ - 1)}

therefore

d e_{i n t} = \frac{d P}{ρ (γ - 1)} - \frac{P d ρ}{ρ^{2} (γ - 1)}

substitute into second law:

d S = \frac{1}{T} d e_{i n t} - \frac{1}{T ρ^{2}} P d ρ

d S = \frac{1}{T} (\frac{d P}{ρ (γ - 1)} - \frac{P d ρ}{ρ^{2} (γ - 1)}) - \frac{1}{T ρ^{2}} P d ρ

d S = \frac{1}{T ρ (γ - 1)} d P - \frac{P}{T ρ^{2} (γ - 1)} d ρ - \frac{1}{T ρ^{2}} P d ρ

d S = \frac{1}{T ρ (γ - 1)} d P - \frac{γ}{γ - 1} \frac{P}{T ρ^{2}} d ρ

second law partials for polytropic ideal gas:

\partial_{P} S = \frac{1}{T ρ (γ - 1)}

\partial_{ρ} S = - \frac{γ}{γ - 1} \frac{P}{T ρ^{2}}

substitute

T = \frac{e_{i n t}}{c_{v}} = \frac{P}{(γ - 1) ρ c_{v}}

our partials of

S

become:

\frac{\partial S}{\partial P} = \frac{c_{v}}{P}

\frac{\partial S}{\partial ρ} = - γ \frac{c_{v}}{ρ}

solve to find the entropy function:

d S = \frac{c_{v}}{P} d P - γ \frac{c_{v}}{ρ} d ρ

S = S_{0} + c_{v} (l n (P) - l n (P_{0})) - γ c_{v} (l n (ρ) - l n (ρ_{0}))

S = S_{0} + c_{v} l n (\frac{P}{P_{0}} (\frac{ρ_{0}}{ρ})^{γ})

The entropy function should be constant (i.e.

\frac{d S}{d t} = 0

except in the presence of shock waves.
Notice that velocity does not appear in the entropy function. Nor does it appear later in the entropy flux function

ρ S

. This might be why the entropy flux function gradient with respect to primitives matches the diagonalized quasilinear solution, which is equal to the acoustic matrix plus identity times

v_{x}

. This

v_{i}

variables might be distinct of the entropy flux, but they are also distinct of what its gradient computes.

Characteristics

primitives:

W_{I} =↓ I [\begin{matrix} ρ \\ v_{i} \\ P \end{matrix}]

conservatives:

U_{I} =↓ I [\begin{matrix} ρ \\ ρ v_{i} \\ E_{t o t a l} \end{matrix}]

for

E_{t o t a l} = ρ e_{t o t a l}

= total energy density

e_{t o t a l} = e_{i n t} + e_{k i n} = e_{i n t} + \frac{1}{2} v^{2}

= total specific energy

Derivative of conservative wrt primitive:

\frac{\partial U_{I}}{\partial W_{J}} =↓ I \overset{\to J}{[\begin{matrix} 1 & 0 & 0 \\ v_{i} & ρ δ_{i j} & 0 \\ \frac{1}{2} v^{2} & ρ v_{j} & \frac{1}{γ - 1} \end{matrix}]}

Derivative of primitives wrt conservatives:

\frac{\partial W_{J}}{\partial U_{I}} =↓ J \overset{\to I}{[\begin{matrix} 1 & 0 & 0 \\ - \frac{v_{j}}{ρ} & δ_{i j} \frac{1}{ρ} & 0 \\ \frac{1}{2} (γ - 1) v^{2} & - (γ - 1) v_{i} & γ - 1 \end{matrix}]}

Flux vector:

F_{I j} =↓ I [\begin{matrix} ρ v_{j} \\ ρ v_{i} v_{j} + δ_{i j} P \\ H_{t o t a l} v_{j} \end{matrix}]

for

H_{t o t a l} = ρ h_{t o t a l}

= total entropy density

h_{t o t a l} = e_{t o t a l} + P

= total specific entropy

Derivative of flux vector with respect to primitive variables:

\frac{\partial F_{I j}}{\partial W_{K}} =↓ I \overset{\to K}{[\begin{matrix} v_{j} & ρ δ_{j k} & 0 \\ v_{i} v_{j} & ρ (δ_{i k} v_{j} + v_{i} δ_{j k}) & δ_{i j} \\ \frac{1}{2} v^{2} v_{j} & ρ v_{j} v_{k} + H_{t o t a l} δ_{j k} & \frac{γ}{γ - 1} v_{j} \end{matrix}]}

Derivative of flux vector with respect to conservative variables:

\frac{\partial F_{I j}}{\partial U_{K}} = \frac{\partial F_{I j}}{\partial W_{L}} \frac{\partial W_{L}}{\partial U_{K}}

=↓ I \overset{\to L}{[\begin{matrix} v_{j} & ρ δ_{j l} & 0 \\ v_{i} v_{j} & ρ (δ_{i l} v_{j} + v_{i} δ_{j l}) & δ_{i j} \\ \frac{1}{2} v^{2} v_{j} & ρ v_{j} v_{l} + (E_{h y d r o} + P) δ_{j l} & \frac{γ}{γ - 1} v_{j} \end{matrix}]} \cdot ↓ L \overset{\to K}{[\begin{matrix} 1 & 0 & 0 \\ - \frac{1}{ρ} v_{l} & \frac{1}{ρ} δ_{k l} & 0 \\ \frac{1}{2} (γ - 1) v^{2} & - (γ - 1) v_{k} & γ - 1 \end{matrix}]}

=↓ I \overset{\to K}{[\begin{matrix} 0 & δ_{j k} & 0 \\ - v_{i} v_{j} + \frac{1}{2} δ_{i j} (γ - 1) v^{2} & δ_{i k} v_{j} + δ_{j k} v_{i} - δ_{i j} (γ - 1) v_{k} & δ_{i j} (γ - 1) \\ v_{j} (\frac{1}{2} (γ - 1) v^{2} - h_{t o t a l}) & - (γ - 1) v_{j} v_{k} + δ_{j k} h_{t o t a l} & γ v_{j} \end{matrix}]}

Hyperbolic conservation law:

\frac{\partial U_{I}}{\partial t} + \frac{\partial F_{I j}}{\partial x_{J}} = 0

with respect to conservative variables:

\frac{\partial U_{I}}{\partial t} + \frac{\partial F_{I j}}{\partial U_{K}} \frac{\partial U_{K}}{\partial x_{j}} = 0

Rewritten in terms of primitives:

\frac{\partial U_{I}}{\partial W_{K}} \frac{\partial W_{K}}{\partial t} + \frac{\partial F_{I j}}{\partial U_{K}} \frac{\partial U_{K}}{\partial x_{j}} = 0

\frac{\partial W_{L}}{\partial U_{I}} \frac{\partial U_{I}}{\partial W_{K}} \frac{\partial W_{K}}{\partial t} + \frac{\partial W_{L}}{\partial U_{I}} \frac{\partial F_{I j}}{\partial U_{K}} \frac{\partial U_{K}}{\partial x_{j}} = 0

δ_{L K} \frac{\partial W_{K}}{\partial t} + \frac{\partial W_{L}}{\partial U_{I}} \frac{\partial F_{I j}}{\partial U_{K}} \frac{\partial U_{K}}{\partial x_{j}} = 0

\frac{\partial W_{L}}{\partial t} + \frac{\partial W_{L}}{\partial U_{I}} \frac{\partial F_{I j}}{\partial U_{M}} \frac{\partial U_{M}}{\partial W_{K}} \frac{\partial W_{K}}{\partial x_{j}} = 0

reindex

\frac{\partial W_{I}}{\partial t} + \frac{\partial W_{I}}{\partial U_{L}} \frac{\partial F_{L j}}{\partial U_{M}} \frac{\partial U_{M}}{\partial W_{K}} \frac{\partial W_{K}}{\partial x_{j}} = 0

Therefore the equivalent flux matrix for the primitive variables - the quasilinear flux matrix - is given by:

\frac{\partial W_{I}}{\partial U_{L}} \frac{\partial F_{L j}}{\partial U_{M}} \frac{\partial U_{M}}{\partial W_{K}} = \frac{\partial W_{I}}{\partial U_{L}} \frac{\partial F_{L j}}{\partial W_{K}}

=↓ I \overset{\to L}{[\begin{matrix} 1 & 0 & 0 \\ - \frac{v_{i}}{ρ} & δ_{i l} \frac{1}{ρ} & 0 \\ \frac{1}{2} (γ - 1) v^{2} & - (γ - 1) v_{l} & γ - 1 \end{matrix}]} \cdot \frac{\partial F_{L j}}{\partial W_{K}}

=↓ I \overset{\to L}{[\begin{matrix} 1 & 0 & 0 \\ - \frac{v_{i}}{ρ} & δ_{i l} \frac{1}{ρ} & 0 \\ \frac{1}{2} (γ - 1) v^{2} & - (γ - 1) v_{l} & γ - 1 \end{matrix}]} \cdot ↓ L \overset{\to K}{[\begin{matrix} v_{j} & ρ δ_{j k} & 0 \\ v_{l} v_{j} & ρ (δ_{l k} v_{j} + v_{l} δ_{j k}) & δ_{l j} \\ \frac{1}{2} v^{2} v_{j} & ρ v_{j} v_{k} + H_{t o t a l} δ_{j k} & \frac{γ}{γ - 1} v_{j} \end{matrix}]}

=↓ I \overset{\to K}{[\begin{matrix} v_{j} & ρ δ_{j k} & 0 \\ 0 & δ_{i k} v_{j} & δ_{i j} \frac{1}{ρ} \\ 0 & γ P δ_{j k} & v_{j} \end{matrix}]}

=↓ I \overset{\to K}{[\begin{matrix} 0 & ρ δ_{j k} & 0 \\ 0 & 0 & δ_{i j} \frac{1}{ρ} \\ 0 & γ P δ_{j k} & 0 \end{matrix}]} + δ_{I K} v_{j}

= A_{I j K} + δ_{I K} v_{j}

for

A_{I j K} =↓ I \overset{\to K}{[\begin{matrix} 0 & ρ δ_{j k} & 0 \\ 0 & 0 & δ_{i j} \frac{1}{ρ} \\ 0 & γ P δ_{j k} & 0 \end{matrix}]} =

the acoustic matrix in direction

x_{j}

Let

c_{s n d}^{2} = \frac{γ P}{ρ}

be the speed of sound.

Right eigenvectors of

A_{I j K}

A_{I j K} y_{K} = λ y_{K}

↓ I \overset{\to K}{[\begin{matrix} 0 & ρ δ_{j k} & 0 \\ 0 & 0 & δ_{i j} \frac{1}{ρ} \\ 0 & γ P δ_{j k} & 0 \end{matrix}]} \cdot y_{K}

has the following solutions:

{\begin{matrix} ρ δ_{j k} y_{k} = λ y_{ρ} \\ \frac{1}{ρ} δ_{i j} y_{P} = λ y_{i} \\ γ P δ_{j k} y_{k} = λ y_{P} \end{matrix}}

{\begin{matrix} y_{j} = λ y_{ρ} / ρ \\ y_{i} = δ_{i j} y_{P} / (ρ λ) \\ y_{j} = λ y_{P} / (γ P) \end{matrix}}

combining

y_{j}

of the 1st and 3rd constraints:

λ y_{ρ} / ρ = λ y_{P} / (γ P)

divide by

λ

(γ P) / ρ = y_{P} / y_{ρ}

Let

y_{ρ} = 1

Therefore

y_{P} = \frac{γ P}{ρ} = c_{s n d}^{2}

substitute

y_{P}

into the 2nd constraint:

y_{i} = δ_{i j} \frac{γ P}{ρ^{2} λ}

equate this with the 2nd constraint:

y_{j} = \frac{γ P}{ρ^{2} λ} = λ / ρ

solve for

λ

λ^{2} = \frac{γ P}{ρ} = c_{s n d}^{2}

λ = \pm c_{s n d}

use

λ

and

y_{i} = δ_{i j} λ / ρ

to solve for

y_{i}

y_{i} = \pm δ_{i j} \frac{c_{s n d}}{ρ}

Then there is

λ = 0

(TODO show where this comes from)

{\begin{matrix} ρ δ_{j k} y_{k} = 0 \\ \frac{1}{ρ} δ_{i j} y_{P} = 0 \\ γ P δ_{j k} y_{k} = 0 \end{matrix}}

This is true for

y_{j} = 0

by the 1st and 3rd constraint, and

y_{P} = 0

by the 2nd constraint
Therefore we are left with three degrees of freedom: one of

y_{ρ} = 1

, and two of

y_{j^{'}} = 1

for

j^{'} \neq j

Put these together and we get:

A_{I j K} y_{K L} = Λ_{I K} y_{K L}

For

Λ_{I K} = δ_{I K} λ_{K}

is the diagonal matrix of eigenvalues
and

y_{K L}

are the collection of eigenvectors

y_{K}

, for the index

L

denoting the distinct eigenvector.
Written out:

↓ I \overset{\to K}{[\begin{matrix} 0 & ρ δ_{j k} & 0 \\ 0 & 0 & \frac{1}{ρ} δ_{i j} \\ 0 & γ P δ_{j k} & 0 \end{matrix}]} \cdot ↓ K \overset{\to L}{[\begin{matrix} 1 & 1 & 0 & 1 \\ - δ_{i j} \frac{c_{s n d}}{ρ} & 0 & δ_{j^{'} j} & δ_{i j} \frac{c_{s n d}}{ρ} \\ c_{s n d}^{2} & 0 & 0 & c_{s n d}^{2} \end{matrix}]} =↓ I \overset{\to K}{[\begin{matrix} - c_{s n d} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & c_{s n d} \end{matrix}]} \cdot ↓ K \overset{\to L}{[\begin{matrix} 1 & 1 & 0 & 1 \\ - δ_{i j} \frac{c_{s n d}}{ρ} & 0 & δ_{j^{'} j} & δ_{i j} \frac{c_{s n d}}{ρ} \\ c_{s n d}^{2} & 0 & 0 & c_{s n d}^{2} \end{matrix}]}

Now we can find the eigenvalues of the change in flux with respect to change in conservative as:

= \frac{\partial F_{I j}}{\partial W_{L}} \frac{\partial W_{L}}{\partial U_{K}}

by chain rule

= δ_{I N} \frac{\partial F_{N j}}{\partial W_{L}} \frac{\partial W_{L}}{\partial U_{K}}

by inserting a delta

= \frac{\partial U_{I}}{\partial W_{M}} \frac{\partial W_{M}}{\partial U_{N}} \frac{\partial F_{N j}}{\partial W_{L}} \frac{\partial W_{L}}{\partial U_{K}}

by replacing the delta with a change in coordinates and its inverse
Now we can replace the

\frac{\partial W_{M}}{\partial U_{N}} \frac{\partial F_{N j}}{\partial W_{L}}

with the acoustic matrix

A_{N j l} + δ_{N L} v_{j}

= \frac{\partial U_{I}}{\partial W_{M}} (A_{N j L} + δ_{N L} v_{j}) \frac{\partial W_{L}}{\partial U_{K}}

So the eigenvectors of

\frac{\partial F_{I j}}{\partial U_{K}}

are equal to those of

A_{I j K}

plus

v_{j}

(why? because

A_{I j K}

is traceless?).
And (only if the eigenvectors of

A_{I j K}

are equal to those of

δ_{I K} v_{j}

. why again?) the right eigenvectors of

\frac{\partial F_{I j}}{\partial U_{K}}

are equal to

\frac{\partial U_{I}}{\partial W_{K}} y_{K L}

Entropy Function

Entropy recap:

S = S_{0} + c_{v} l n (\frac{P}{P_{0}} (\frac{ρ_{0}}{ρ})^{γ}

Derivative of

S

wrt primitives:

\partial_{ρ} S = - γ c_{v} / ρ

\partial_{v_{i}} S = 0

\partial_{P} S = c_{v} / P

\partial_{I} S = \overset{\to I}{[\begin{matrix} - \frac{γ c_{v}}{ρ} & 0 & \frac{c_{v}}{P} \end{matrix}]}

Derivative of densitised entropy wrt primitives:

\partial_{ρ} (ρ S) = S - γ c_{v}

\partial_{v_{i}} (ρ S) = 0

\partial_{P} (ρ S) = c_{v} ρ / P

\partial_{I} S = \overset{\to I}{[\begin{matrix} S - γ c_{v} & 0 & \frac{c_{v} ρ}{P} \end{matrix}]}

...transformed by the transpose quasilinear flux matrix...

\partial_{I} (ρ S) \frac{\partial W_{I}}{\partial U_{L}} \frac{\partial F_{L j}}{\partial W_{K}}

= \overset{\to I}{[\begin{matrix} S - γ c_{v} & 0 & \frac{c_{v} ρ}{P} \end{matrix}]} \cdot ↓ I \overset{\to K}{[\begin{matrix} v_{j} & ρ δ_{j k} & 0 \\ 0 & δ_{i k} v_{j} & δ_{i j} \frac{1}{ρ} \\ 0 & γ P δ_{j k} & v_{j} \end{matrix}]}

= \overset{\to K}{[\begin{matrix} v_{j} (S - γ c_{v}) & δ_{j k} ρ S & \frac{v_{j} c_{v} ρ}{P} \end{matrix}]}

= ρ v_{j} \partial_{I} S + S [\begin{matrix} v_{j} & δ_{j k} ρ & 0 \end{matrix}]

= ρ v_{j} \partial_{I} S + S \partial_{I} (ρ v_{j})

= \partial_{I} (S ρ v_{j})

So the gradient of the densitized entropy function, times the quasilinear flux matrix, is equal to the gradient of the combined densitized entropy function times the velocity in the flux direction.
Is that the conditions for the definition of the entropy flux?

Second derivative of densitized entropy with respect to primitives:

(ρ S)_{, ρ ρ} = S_{, ρ} = - \frac{γ c_{v}}{ρ}

(ρ S)_{, ρ P} = S_{, P} = \frac{c_{v}}{P}

(ρ S)_{, v_{i} K} = 0

(ρ S)_{, P P} = (\frac{c_{v} ρ}{P})_{, P} = - \frac{c_{v} ρ}{P^{2}}

(ρ S)_{I K} = c_{v} \cdot ↓ I \overset{\to K}{[\begin{matrix} - \frac{γ}{ρ} & 0 & \frac{1}{P} \\ 0 & 0 & 0 \\ \frac{1}{P} & 0 & - \frac{ρ}{P^{2}} \end{matrix}]}

eigenvalues of second derivative of entropy with respect to primitives:

λ = 0

times 3
and

(- c_{v} \frac{γ}{ρ} - λ) (- c_{v} \frac{ρ}{P^{2}} - λ) - \frac{{c_{v}}^{2}}{P^{2}} = 0

λ^{2} + c_{v} (\frac{γ}{ρ} + \frac{ρ}{P^{2}}) λ + {c_{v}}^{2} \frac{γ - 1}{P^{2}} = 0

λ = \frac{1}{2} {c_{v}}^{2} (- (\frac{γ}{ρ} + \frac{ρ}{P^{2}}) \pm \sqrt{(\frac{γ}{ρ} + \frac{ρ}{P^{2}})^{2} - 4 \frac{γ - 1}{P^{2}}})

λ = - \frac{1}{2} {c_{v}}^{2} (\frac{γ}{ρ} + \frac{ρ}{P^{2}} \mp \sqrt{\frac{γ^{2}}{ρ^{2}} + 2 \frac{γ}{P^{2}} + \frac{ρ^{2}}{P^{4}} - 4 \frac{γ - 1}{P^{2}}})

λ = - \frac{1}{2} {c_{v}}^{2} (\frac{γ}{ρ} + \frac{ρ}{P^{2}} \mp \sqrt{\frac{γ^{2}}{ρ^{2}} - 2 \frac{γ}{P^{2}} + \frac{ρ^{2}}{P^{4}} + \frac{4}{P^{2}}})