Euler equations in primitive form for a real gas

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For an ideal gas, it is relatively easy to express the Euler equations in primitive form (variables $rho$, $u$, $p$), starting from their expression in conservative variables ($rho$, $rho u$, $E$).

I did not find any example of such derivation for a general real gas, governed by any equation of state. Is it possible to express the Euler equations in primitive form for any (unknown) real gas (involving the speed of sound somewhere)?

thermodynamics fluid-dynamics

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edited Jul 31 at 1:34

Qmechanic♦

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For an ideal gas, it is relatively easy to express the Euler equations in primitive form (variables $rho$, $u$, $p$), starting from their expression in conservative variables ($rho$, $rho u$, $E$).

I did not find any example of such derivation for a general real gas, governed by any equation of state. Is it possible to express the Euler equations in primitive form for any (unknown) real gas (involving the speed of sound somewhere)?

thermodynamics fluid-dynamics

share|cite|improve this question

edited Jul 31 at 1:34

Qmechanic♦

95.7k12162999

asked Jul 30 at 10:34

seb007

migrated from math.stackexchange.com Jul 31 at 1:07

This question came from our site for people studying math at any level and professionals in related fields.

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

For an ideal gas, it is relatively easy to express the Euler equations in primitive form (variables $rho$, $u$, $p$), starting from their expression in conservative variables ($rho$, $rho u$, $E$).

I did not find any example of such derivation for a general real gas, governed by any equation of state. Is it possible to express the Euler equations in primitive form for any (unknown) real gas (involving the speed of sound somewhere)?

thermodynamics fluid-dynamics

share|cite|improve this question

edited Jul 31 at 1:34

Qmechanic♦

95.7k12162999

asked Jul 30 at 10:34

seb007

For an ideal gas, it is relatively easy to express the Euler equations in primitive form (variables $rho$, $u$, $p$), starting from their expression in conservative variables ($rho$, $rho u$, $E$).

I did not find any example of such derivation for a general real gas, governed by any equation of state. Is it possible to express the Euler equations in primitive form for any (unknown) real gas (involving the speed of sound somewhere)?

thermodynamics fluid-dynamics

share|cite|improve this question

edited Jul 31 at 1:34

Qmechanic♦

95.7k12162999

asked Jul 30 at 10:34

seb007

share|cite|improve this question

share|cite|improve this question

edited Jul 31 at 1:34

Qmechanic♦

95.7k12162999

edited Jul 31 at 1:34

Qmechanic♦

95.7k12162999

edited Jul 31 at 1:34

Qmechanic♦

95.7k12162999

95.7k12162999

asked Jul 30 at 10:34

seb007

asked Jul 30 at 10:34

seb007

asked Jul 30 at 10:34

seb007

migrated from math.stackexchange.com Jul 31 at 1:07

This question came from our site for people studying math at any level and professionals in related fields.

migrated from math.stackexchange.com Jul 31 at 1:07

This question came from our site for people studying math at any level and professionals in related fields.

add a comment | 

add a comment | 

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Yes it can always be done. I assume you can write the general case in conservation form. so you already have one primitive variable, Then

$$u_t=rho^-1((rho u)_t-urho_t)$$

and$$i_t=rho^-1[(rho i+rho u^2/2)_t-(i+u^2/2)rho_t-rho uu_t]$$

where i is specific internal energy. Generalization to more dimensions is obvious. You should see some simplification.

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answered Jul 30 at 11:30

Philip Roe

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1 Answer
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Yes it can always be done. I assume you can write the general case in conservation form. so you already have one primitive variable, Then

$$u_t=rho^-1((rho u)_t-urho_t)$$

and$$i_t=rho^-1[(rho i+rho u^2/2)_t-(i+u^2/2)rho_t-rho uu_t]$$

where i is specific internal energy. Generalization to more dimensions is obvious. You should see some simplification.

share|cite|improve this answer

answered Jul 30 at 11:30

Philip Roe

45515

add a comment | 

up vote
0
down vote

Yes it can always be done. I assume you can write the general case in conservation form. so you already have one primitive variable, Then

$$u_t=rho^-1((rho u)_t-urho_t)$$

and$$i_t=rho^-1[(rho i+rho u^2/2)_t-(i+u^2/2)rho_t-rho uu_t]$$

where i is specific internal energy. Generalization to more dimensions is obvious. You should see some simplification.

share|cite|improve this answer

answered Jul 30 at 11:30

Philip Roe

45515

add a comment | 

up vote
0
down vote

up vote
0
down vote

Yes it can always be done. I assume you can write the general case in conservation form. so you already have one primitive variable, Then

$$u_t=rho^-1((rho u)_t-urho_t)$$

and$$i_t=rho^-1[(rho i+rho u^2/2)_t-(i+u^2/2)rho_t-rho uu_t]$$

where i is specific internal energy. Generalization to more dimensions is obvious. You should see some simplification.

share|cite|improve this answer

answered Jul 30 at 11:30

Philip Roe

45515

Yes it can always be done. I assume you can write the general case in conservation form. so you already have one primitive variable, Then

$$u_t=rho^-1((rho u)_t-urho_t)$$

and$$i_t=rho^-1[(rho i+rho u^2/2)_t-(i+u^2/2)rho_t-rho uu_t]$$

where i is specific internal energy. Generalization to more dimensions is obvious. You should see some simplification.

share|cite|improve this answer

answered Jul 30 at 11:30

Philip Roe

45515

share|cite|improve this answer

share|cite|improve this answer

answered Jul 30 at 11:30

Philip Roe

45515

answered Jul 30 at 11:30

Philip Roe

45515

answered Jul 30 at 11:30

Philip Roe

45515

45515

add a comment | 

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