Compressible flow

Compressible flow allows density to change materially and is required for finite-amplitude pressure waves and shocks.

A self-similar blast wave is an expanding shock wave whose post-shock profiles retain their shape in a rescaled radius or distance. A fixed explosion energy, ambient mass density and geometry set the front scaling through conservation of total energy. The dimensionless coefficient requires the profiles and an energy-normalization convention.

For a planar adiabatic explosion with energy per unit area $E$ in a uniform medium of mass density ${\rho }_0$, the conserved energy has form $E=I{\rho }_{0}Z{\dot{Z}}^2$, with a finite positive integral $I$ of the similarity profiles. Hence $Z=C(E/{\rho }_0{)}^{1/3}{t}^{2/3}$. If $E$ is shared by two fronts, its fraction assigned to either side is absorbed into $C$.

A polytropic flow uses a polytropic equation of state. When $K$ and the adiabatic exponent are constant, a smooth streamline has specific enthalpy $h=\gamma K{\rho }^{\gamma -1}/(\gamma -1)$ and adiabatic sound speed $c_s^2=\gamma K{\rho }^{\gamma -1}$. Spatially varying entropy can make $K$ depend on streamline; the constant-$K$ spherical wind is a special case.

A smooth radial compressible flow with a fixed adiabatic exponent conserves $r^2\rho u$ and $p/{\rho }^{\gamma }$. In a prescribed Newtonian gravitational potential, its Bernoulli function is $u^2/2+c^2/(\gamma -1)+\Phi$, with $c$ the adiabatic sound speed. Eliminating the density derivative yields $(u-c^2/u)u^{\prime }=2c^2/r-{\Phi }^{\prime }$.

A smooth transonic branch in a power-law gravitational potential has $u_{\ast }^2=c_{\ast }^2=A\beta /(2r_{\ast }^{\beta })$ at its sonic point. Substitution into the Bernoulli function gives $\mathcal{B}=c_{\ast }^2[1/(\gamma -1)+1/2-2/\beta ]$. The local sonic-point slope discriminant distinguishes a genuine crossing from a degenerate everywhere-sonic scaling solution.

With reservoir density ${\rho }_{\infty }$ and adiabatic sound speed $c_{\infty }$, put $q=2/\beta -1/2$ and $h=1-q(\gamma -1)>0$. The selected mass accretion rate is $4\pi {\rho }_{\infty }(A\beta /2{)}^{2/\beta }{c}_{\infty }^{1-4/\beta }{h}^{q-1/(\gamma -1)}$. It follows by evaluating the conserved spherical flux at the sonic point.

The dimensionless accretion factor $(1-q\delta {)}^{q-1/\delta }$ tends to $e^q$ as $\delta \to 0$. For $q>0$, it tends to $1$ as $\delta \uparrow 1/q$, although the sonic point moves to the origin. Writing the latter logarithm as $-qh\ln h/(1-h)$ makes the finite limit transparent. These limits connect transonic spherical accretion rate in a power-law potential to the isothermal equation of state and the limiting critical adiabatic exponent.

For $0<\beta <4$, a nondegenerate transonic spherical flow in a power-law potential connected to infinity requires $\gamma <(4+\beta )/(4-\beta )$. For $\beta \ge 4$, every finite $\gamma >1$ satisfies the local crossing condition; there is no finite upper bound. The uniformly valid inequality is $2\beta -(4-\beta )(\gamma -1)>0$.

An isothermal shock joins two states at the same temperature and isothermal sound speed. Mass and momentum conservation imply $u_1{u}_2=c_s^2$ for a nontrivial planar jump, so the compression ratio is the upstream isothermal Mach number squared. Heat exchange replaces adiabatic energy conservation across the gas alone.

The specific enthalpy is equal on both sides of an isothermal perfect-gas shock. The energy removed per unit area per unit time is $j(u_1^2-u_2^2)/2$, with $j>0$ the normal mass flux. Net cooling therefore selects compression shocks; an expansion jump requires heating.

For an inviscid fluid with specific internal energy $e$, let $E=\rho (e+|\mathbf{u}{|}^2/2)$. In a prescribed potential $\Phi$, ${\partial }_{t}E+\nabla \cdot [(E+p)\mathbf{u}]=-\rho \mathbf{u}\cdot \nabla \Phi +Q$, where $Q$ is external heat supply. A static gravitational potential can instead be included in the conserved energy.

For steady spherical isentropic flow with $p=K{\rho }^{3/2}$ around a point mass, put $j=r^2\rho v>0$ and $\mathcal{E}=v^2/2+2c^2-GM/r>0$. Eliminating the mass density gives $c^2=[j(3K/2{)}^2{]}^{2/5}{r}^{-4/5}{\mathcal{M}}^{-2/5}$. With $r_s=GM/(4\mathcal{E})$, $x=(r/r_s{)}^{1/5}$, $y={\mathcal{M}}^{2/5}$ and $\lambda =[j(3K/2{)}^2{]}^{2/5}/(2\mathcal{E}{r}_s^{4/5})$, the Bernoulli equation reduces to $\lambda F(y)=F(x)$, where $F(t)=t^4+4/t$. Since $F$ has its unique minimum at $t=1$, a regular sonic point requires $\lambda =1$ and $r=r_s$. One transonic branch is $y=x$, which has $v=\sqrt{2\mathcal{E}}$, $\rho \propto r^{-2}$ and $p\propto r^{-3}$. The decreasing branch has inner free fall and an outer nearly static reservoir, giving Bondi accretion for inward fluid flow. A nonradiative stationary shock wave keeps $j,\mathcal{E},r_s$ fixed but raises $K$, so $\lambda$ increases by the factor $(K_2/K_1{)}^{4/5}$. The inward transonic solution is the spherical $n=2$, $\gamma =3/2$ specialization of transonic accretion in a power-law tube.

The speed of sound is the propagation speed of a small pressure disturbance through a medium.

The isothermal sound speed is $a_i^2=(\partial p/\partial \rho {)}_T$. For an ideal gas it is $a_i^2=p/\rho =\mathcal{R}T/\mu$.

The adiabatic sound speed is $v_s^2=(\partial p/\partial \rho {)}_s$. For a perfect gas it is $v_s^2=\gamma p/\rho$.

An isentropic flow has constant entropy along the fluid and contains no entropy-producing shocks or dissipation. For a perfect gas, $p/{\rho }^{\gamma }$ is constant and $c_s^2=\gamma p/\rho \propto {\rho }^{\gamma -1}$.

In one spatial dimension, inviscid isentropic mass and momentum conservation are

A homentropic flow has one spatially uniform entropy value throughout the fluid. It is therefore isentropic along every fluid trajectory, and its pressure can be treated as a single-valued function of density.

A globally isothermal fluid has spatially and temporally constant sound speed $c_s$ and pressure $p=c_s^2\rho$. Maintaining this relation generally requires heat exchange rather than adiabatic evolution.

A sonic point is a point at which the flow speed equals the local sound speed, so the Mach number is one.

For spherically symmetric adiabatic flow in a power-law gravitational potential, let $X=(r_{\ast }/u_{\ast })u_{\ast }^{\prime }$ and $\delta =\gamma -1$. Differentiation at a sonic point gives $(2+\delta )X^2+4\delta X+4\delta -2\beta =0$. Its discriminant is $8[2\beta -(4-\beta )\delta ]$; a positive value gives the two local crossing directions.

For a steady polytropic streamline with Bernoulli function $C_B$, the critical speed is the sound speed at which the scalar mass flux $\rho u$ is maximal. It satisfies

A transonic branch is a solution that passes continuously between subsonic and supersonic flow through a regular sonic point.

Consider steady isentropic flow toward a Newtonian gravitational potential $-GM/r$ through a tube of cross-sectional area $A(r)=C{r}^n$. Write $v>0$ for the inward speed and use a polytropic equation of state with specific-heat ratio $\gamma >1$. Mass conservation and the Euler equations for an inviscid fluid givewhere $c_s$ is the adiabatic sound speed. A regular sonic point therefore has $v_s=c_s$ and $r_s=GM/(n{c}_s^2)$. The Bernoulli equation, matched to a nearly stationary reservoir with sound speed $c_0$, givesFor $n>1/2$, a finite positive sonic point requires $1<\gamma <(2n+1)/(2n-1)$. Differentiating the flow equation at that point, with $x=r_s{v}_s^{\prime }/c_s$, givesIts discriminant is $4n[(2n+1)-(2n-1)\gamma ]$, so the same bound permits real regular slopes. The branch on which the Mach number rises inward selects the negative sign inFor a spherical tube $n=2$, this recovers the $5/3$ threshold of Bondi accretion; for a dipolar flux-tube area, $n=3$ gives $7/5$. The tube approximation must remain valid between the reservoir matching region and the accretor.

The Mach number is the ratio of flow speed to local sound speed.

The radial Mach number is the source velocity projected toward the observer, divided by the sound speed. It enters the moving-surface retarded Jacobian through $d|x-y(\tau )|/d\tau =-\hat{r}\cdot v$. The sign is directional: a source moving toward the observer has positive $M_r$ and compressed arrival times.

A subsonic flow has Mach number less than one, so its speed is below the local sound speed.

A supersonic flow has Mach number greater than one, so the flow speed exceeds the local sound speed.

For one-dimensional homentropic ideal-gas flow,is constant along the characteristic $dx/dt=u\pm c$.

For a barotropic pressure law with $c^2=dp/d\rho$, the invariantsobey $[{\partial }_t+(u\pm c){\partial }_x]R_{\pm }=0$.

A simple wave is a region of a hyperbolic flow in which one Riemann invariant is constant throughout, leaving the other invariant to determine all fluid variables.

A characteristic field is linearly degenerate if the directional derivative of its speed along its right eigenvector vanishes. Its simple waves have constant propagation speed along the state curve, so they do not steepen by amplitude-dependent transport.

An ideal-MHD simple wave traces a one-parameter state-space curve whose tangent is a right eigenvector of the characteristic matrix. Its parameter obeys $q_t+v(q)q_x=0$ and its characteristic speed obeys the Inviscid Burgers equation. Nonlinear Alfvén waves are a linearly degenerate example.

For a one-dimensional velocity perpendicular to a transverse magnetic field, $c_f^2=\gamma p/\rho +B^2/({\mu }_0\rho )$. The primitive-variable characteristic matrix has two advective eigenvalues and the two speeds $u\pm c_f$. Integrating the fast eigenvectors keeps $p/{\rho }^{\gamma }$ and $B/\rho$ constant. The state parameter obeys ${\rho }_t+(u\pm c_f){\rho }_x=0$, so the characteristic speed satisfies the Inviscid Burgers equation.

For $u_t+(c_0+bu)u_x=-\alpha u$, $b,\alpha >0$, the characteristic curves are $u=e^{-\alpha t}{u}_0(\xi )$ and $x=c_{0}t+\xi +(b/\alpha )(1-e^{-\alpha t})u_0(\xi )$. If $m=\max [-u_0^{\prime }]>0$, finite-time gradient blow-up occurs precisely for $\alpha <bm$, at $t_s=-{\alpha }^{-1}\log (1-\alpha /(bm))$. Equality $\alpha =bm$ still gives a positive Jacobian at each finite time; only the limiting map degenerates.

For a piston receding from initially stationary homentropic gas, the right-moving simple wave has $R_{-}=0$, so $c=c_0+(\gamma -1)u/2$. A $C_{+}$ characteristic emitted by the piston at time $s$ carries $u={\dot{X}}_p(s)$ and is the straight line

The gas state at a receding piston reaches vacuum when ${\dot{X}}_p=-2c_0/(\gamma -1)$, because then the sound speed and mass density both vanish. If the piston subsequently recedes faster, a vacuum gap opens between it and the gas boundary, which continues with velocity $-2c_0/(\gamma -1)$.

For $X_p(t)=-t^{1+\alpha }/(1+\alpha )$ with $0<\alpha \ll 1$, a point on the ray $x=(c_0-a)t$, $0<a<(\gamma +1)/2$, receives the characteristic emitted at the exponentially small timeAlthough $s\to 0$, the piston speed $-s^{\alpha }$ has the finite limit $-2a/(\gamma +1)$, producing the centered-rarefaction profile of the singular limit $\alpha =0$.

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, soThe isentropic relationconverts the velocity disturbance into its nonlinear pressure disturbance.

A smooth compressive simple wave forms a shock when characteristics first intersect. If characteristics are parametrized by their emission time $\tau$, the first shock occurs at the minimum positive time for which $\partial x(\tau ,t)/\partial \tau =0$.

A normal shock is a discontinuity perpendicular to the flow direction across which mass, momentum, and total energy fluxes are conserved.

For a moving level surface, $n=\nabla S/|\nabla S|$, $v_n=-S_t/|\nabla S|$, and the surface delta distribution is ${\delta }_s=|\nabla S|\delta (S)$. Piecewise classical conservation laws acquire a distributional flux defect $[b]\cdot n-v_n[a]$. Vanishing of this defect is the Rankine-Hugoniot condition.

The shock frame is an inertial reference frame in which the shock is stationary. A travelling shock then becomes a steady flow through a fixed discontinuity.

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfyand

For a stationary normal shock in a polytropic perfect gas, the upstream and downstream normal speeds obey $u_1{u}_2=c_{\mathrm{cr}}^2$, where $c_{\mathrm{cr}}$ is the sonic speed on their common Bernoulli streamline.

When the upstream pressure is negligible, a perfect-gas shock has density ratio $(\gamma +1)/(\gamma -1)$. For $\gamma =5/3$, the downstream density is four times the upstream density.

The shock compression ratio is the downstream-to-upstream density ratio. For a strong perfect-gas shock,In the upstream rest frame, the downstream velocity is $2V_s/(\gamma +1)$ and its pressure is $2{\rho }_0{V}_s^2/(\gamma +1)$.

If a piston moves at speed $V$ into stationary perfect gas and drives a shock at speed $U$, then the shock-frame upstream and downstream speeds are $U$ and $U-V$. If $p_2-p_1=\beta p_1$, the jump conditions give

If a shock travels into stationary gas at speed $V$ while the downstream piston and gas travel at $V/3$, then for $\gamma =3$,

Let $P=p_2/p_1$ and $D={\rho }_2/{\rho }_1$ be the pressure and mass density ratios across a normal shock wave in a perfect gas with specific-heat ratio $\gamma$. Eliminating the velocities from the Rankine-Hugoniot conditions for a perfect gas givesAs $P\to \infty$, the ratio tends to the shock compression ratio $(\gamma +1)/(\gamma -1)$.

For the pressure-density Hugoniot relation for a perfect gas, the change in specific entropy is $[s]/c_v=\log P-\gamma \log D(P)$, with $c_v$ the specific heat capacity at constant volume. Its derivative isIt is positive for a compressive normal shock wave with $P>1$. For a weak shock with $P=1+\delta$, integrating the leading term givesThus the entropy production is cubic in the small pressure jump, even though the mass density and temperature changes already appear at first order.

For $N$ equal weak shocks with individual pressure ratio $1+\delta$ and total ratio $P_s=(1+\delta {)}^N$, entropy production in a perfect-gas shock givesAt fixed $P_s>1$, this tends to zero as the compression is divided into increasingly many smaller weak shocks. A single finite normal shock wave instead gives the positive value $\log P_s-\gamma \log D(P_s)$. Gradual compression can consequently approach reversible isentropic flow while an abrupt compression generates finite entropy.

A weak shock has a small relative pressure jump. Its density jump agrees with the reversible adiabatic prediction through second order in the pressure jump, while entropy production first appears at third order.

An oblique shock is inclined relative to the upstream velocity. In an inviscid perfect gas, the normal velocity satisfies the normal-shock jump conditions while the tangential velocity is continuous.

If $p_2/p_1=1+ϵ$ with $0<ϵ\ll 1$ and the upstream Mach number is $M_1>1$, the shock angle is the Mach angle to leading order and the flow deflection isThe normal speed decreases, so the downstream flow turns toward the shock front.

The Mach angle of a supersonic flow with Mach number $M>1$ is $\mu =\arcsin (M^{-1})$. Infinitesimal pressure disturbances lie on the corresponding Mach cone, and a weak oblique shock approaches this angle as its strength tends to zero.