2.2 CompressibilityNiklas Andersson
Chalmers University of Technology
Department of Mechanical Engineering
Division of Fluid Dynamics
If density fluctuations are significant, compressible effects needs to be accounted for, but the question is what significant means. Anderson suggests that fluctuations in the order of 5% (Δ ρ / ρ > 0.05 \Delta \rho/\rho > 0.05 Δ ρ / ρ > 0.05
Let’s try to figure out...
The compressibility for isothermal process is defined as
τ T = ( 1 ρ ) ( ∂ ρ ∂ p ) T \tau_T=\left(\frac{1}{\rho}\right)\left(\frac{\partial \rho}{\partial p}\right)_T τ T = ( ρ 1 ) ( ∂ p ∂ ρ ) T τ T = ( 1 ρ ) ( ∂ ρ ∂ p ) T = { p = ρ R T , T = c o n s t } = ( R T p ) ( ∂ ∂ p ( p R T ) ) T \tau_T=\left(\frac{1}{\rho}\right)\left(\frac{\partial \rho}{\partial p}\right)_T=\left\{p=\rho RT,\ T=const\right\}=\left(\frac{RT}{p}\right)\left(\frac{\partial}{\partial p}\left(\frac{p}{RT}\right)\right)_T τ T = ( ρ 1 ) ( ∂ p ∂ ρ ) T = { p = ρRT , T = co n s t } = ( p RT ) ( ∂ p ∂ ( RT p ) ) T and thus
τ T = 1 p \tau_T=\frac{1}{p} τ T = p 1 We are trying to find an estimate of Δ ρ / ρ \Delta \rho/\rho Δ ρ / ρ Δ ρ / ρ \Delta \rho/\rho Δ ρ / ρ
Δ ρ ρ = Δ p R T R T p = { τ T = 1 p } = τ T Δ p \frac{\Delta \rho}{\rho}=\frac{\Delta p}{RT}\frac{RT}{p}=\left\{\tau_T=\frac{1}{p}\right\}=\tau_T\Delta p ρ Δ ρ = RT Δ p p RT = { τ T = p 1 } = τ T Δ p If we assume that compressible effects are not significant and use Bernoulli’s equation to get an estimate of the pressure fluctuations generated by a flow
Δ p ≈ 1 2 ρ ∞ U ∞ 2 \Delta p\approx \frac{1}{2}\rho_\infty U^2_\infty Δ p ≈ 2 1 ρ ∞ U ∞ 2 With τ T = 1 / p ∞ \tau_T=1/p_\infty τ T = 1/ p ∞ (2.5)
Δ ρ ρ ∞ ≈ 1 p ∞ ( 1 2 ρ ∞ U ∞ 2 ) = { p ∞ = ρ ∞ R T ∞ } = ρ ∞ U ∞ 2 2 ρ ∞ R T ∞ = U ∞ 2 2 R T ∞ \frac{\Delta \rho}{\rho_\infty}\approx \frac{1}{p_\infty}\left(\frac{1}{2}\rho_\infty U^2_\infty\right)=\left\{p_\infty=\rho_\infty R T_\infty\right\}=\frac{\rho_\infty U^2_\infty}{2 \rho_\infty R T_\infty}=\frac{U^2_\infty}{2 R T_\infty} ρ ∞ Δ ρ ≈ p ∞ 1 ( 2 1 ρ ∞ U ∞ 2 ) = { p ∞ = ρ ∞ R T ∞ } = 2 ρ ∞ R T ∞ ρ ∞ U ∞ 2 = 2 R T ∞ U ∞ 2 The speed of sound in the freestream is obtained as a ∞ = γ R T ∞ a_\infty=\sqrt{\gamma RT_\infty} a ∞ = γ R T ∞
Δ ρ ρ ∞ ≈ γ U ∞ 2 2 a ∞ 2 = γ 2 M ∞ 2 \frac{\Delta \rho}{\rho_\infty}\approx\frac{\gamma U^2_\infty}{2 a^2_\infty}=\frac{\gamma}{2}M^2_\infty ρ ∞ Δ ρ ≈ 2 a ∞ 2 γ U ∞ 2 = 2 γ M ∞ 2 For air (γ = 1.4 \gamma=1.4 γ = 1.4 Δ ρ / ρ ∞ < 0.05 \Delta \rho/\rho_\infty<0.05 Δ ρ / ρ ∞ < 0.05 M ∞ < 0.27 M_\infty<0.27 M ∞ < 0.27