Hydrodynamic Viscous Fluid Materials (LAW6)

This law is specifically designed to model liquids and gases.

The equations used to describe the material are:

S i j = 2 ρ v e ˙ i j
p = C 0 + C 1 μ + C 2 μ 2 + C 3 μ 3 + ( C 4 + C 5 μ ) E n

Where,
S i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38D7@
Deviatoric stress tensor
V →
Kinematic viscosity
e ˙ i j
Deviatoric strain rate tensor

The kinematic viscosity V → is related to the dynamic viscosity, η by:

v = η ρ

Perfect Gas Model

To model a perfect gas, all coefficients C0, C1, C2, C3 must be set to equal zero. Also:

C 4 = C 5 = γ − 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4qamaaBaaaleaacaaI0aaabeaakiabg2da9iaadoeadaWgaaWc baGaaGynaaqabaGccqGH9aqpcqaHZoWzcqGHsislcaaIXaaaaa@4180@
E n 0 = P 0 γ − 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaamyramaaBaaaleaacaWGUbGaaGimaaqabaGccqGH9aqpdaWcaaqa aiaadcfadaWgaaWcbaGaaGimaaqabaaakeaacqaHZoWzcqGHsislca aIXaaaaaaa@4183@

A perfect gas allows compressibility and expansion and contraction with a rise in temperature. However, for many situations, especially very slow subsonic flows, an incompressible gas returns accurate and reliable results with less computation.

Incompressible Gas Model

To model an incompressible gas, the coefficients should be set to:

C 0 = C 2 = C 3 = C 4 = C 5 = E 0 =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4qamaaBaaaleaacaaIWaaabeaakiabg2da9iaadoeadaWgaaWc baGaaGOmaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaaiodaaeqaaO Gaeyypa0Jaam4qamaaBaaaleaacaaI0aaabeaakiabg2da9iaadoea daWgaaWcbaGaaGynaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaaic daaeqaaOGaeyypa0JaaGimaaaa@49E7@
C 1 = ρ 0 ⋅ c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4qamaaBaaaleaacaaIXaaabeaakiabg2da9iabeg8aYnaaBaaa leaacaaIWaaabeaakiabgwSixlaadogadaahaaWcbeqaaiaaikdaaa aaaa@4236@

Where, c is the speed of sound.

Incompressibility is achieved via a penalty method. The sound speed is set to at least 10 times the maximum velocity.

This classical assumption is not valid when fluid and structures are coupled. In this case, set the sound speed in the fluid so that the first eigen frequency is at least 10 times higher in the fluid than in the structure.