Shock Waves Steepening of sound waves We have the result that the velocity of a sound wave in an arbitrary reference frame is given by: v u kˆ c s kˆ where u is the velocity of the fluid and k is the wave vector. In one dimension: Consider the velocity profile in a sound wave that is non-linear: + v u+ c s () () Astrophysical Gas Dynamics: Shocks /47
v u + c s u + c s x The velocity of sound is larger at larger velocity amplitudes so that the wave profile evolves in the following way: Astrophysical Gas Dynamics: Shocks /47
u Multiply valued velocity profile x The velocity profile eventually becomes multiply valued and at this point the solution breaks down, requiring the insertion of a discontinuity into the solution. Astrophysical Gas Dynamics: Shocks 3/47
Shocks as discontinuities. Basic approach We have seen that as velocities approach the speed of sound the nonlinearity of the Euler equations forces waves to become steeper and multiple valued. At the point where the velocity profile becomes infinitely steep we intervene and insert a surface of discontinuity into the fluid. Physically this discontinuity represents a region where the fluid variables are rapidly varying and we shall assess later some of the details of this region. Here we look at the consequences of energy and momentum conservation across the surface of discontinuity which we refer to as a shock wave. We develop equations for non-magnetised shocks here and extend this to magnetised shocks in later chapters. Astrophysical Gas Dynamics: Shocks 4/47
Velocity discontinuities in the frame of the shock y v x v Shock Astrophysical Gas Dynamics: Shocks 5/47
The shock relations are best analysed in the frame of the shock (in which its velocity is zero). The coordinate x is normal to the shock; y is in the plane of the pre-shock velocity and is along the shock plane; z is normal to x and y. The general situation depicted above is an oblique shock. When the shock is normal. v v y v y 0 In general many fluid variables, specifically density, pressure and the normal component of the velocity are discontinuous at a shock. Astrophysical Gas Dynamics: Shocks 6/47
. Conservation laws satisfied at the shock Mass flux There is no creation of mass at the shock so that the mass flux into the shock must equal the mass flux emerging from the shock. i.e. v x v x v x 0 where the square brackets refer the jump in a variable across the shock. Astrophysical Gas Dynamics: Shocks 7/47
Momentum Flux Likewise there is no creation of momentum at the shock front and the x and y components of the momentum flux are the same on both sides of the shock. The momentum flux ij v i v j + p ij so that the flux of x-momentum normal to the shock is: (3) p xx v x + v x + p and the flux of y-momentum normal to the shock xy v x v y v x v y (4) (5) Astrophysical Gas Dynamics: Shocks 8/47
Energy Flux The x-component of the energy flux into and out of either side of the shock is F E x v x --v + h (6) so that v x --v + h v x --v + h Astrophysical Gas Dynamics: Shocks 9/47
Summary The above equations all collected into one set are: v x v x v x + p v x v y v x v y + p v x v x --v + h v x --v + h (7) Astrophysical Gas Dynamics: Shocks 0/47
.3 Two types of discontinuity Tangential or contact discontinuity The first type of discontinuity is a tangential discontinuity or contact discontinuity. This occurs when there is no mass flux across the surface, i.e. v x v x 0 For non-zero densities this implies that v x v x 0 When this is the case then the continuity of the y-component of momentum can be satisfied with v y v y Astrophysical Gas Dynamics: Shocks /47 (8) (9) (0)
The xx-component of the momentum flux is continuous if p p This situation is depicted in the following figure: p p p p v y v y v y v y v x 0 v x 0 Astrophysical Gas Dynamics: Shocks /47
Such a situation is unstable to the Kelvin-Helmholtz instability when the difference in velocities is non-zero..4 Shock discontinuity In this case there is a non-zero mass flux across the discontinuity. The continuity of the xy-component of momentum flux implies that v y v y () i.e. the component of velocity tangential to the shock is conserved (i.e. unchanged). Astrophysical Gas Dynamics: Shocks 3/47
Now consider the continuity of energy flux which can be expressed in the form: v x -- v x + v y + h v x -- v x + v y + h () Since v x 0 and v y 0 then the leading factors on the left and right hand sides are identical, so that -- v x v + y + h -- v x + v y + h (3) Astrophysical Gas Dynamics: Shocks 4/47
We can then delete and from this equation since they v y v y are equal. Hence, conservation of energy can be expressed in the form: --v x + h --v x + h (4) Astrophysical Gas Dynamics: Shocks 5/47
Thus, the complete set of equations for shocks can be expressed in the form v x v x j Mass Flux p + v p x + v x x-momentum flux v y v y y-momentum flux h --v + x h + --v x Energy Flux (5).5 The shock adiabat (the Rankine-Hugoniot equations) We now introduce the Specific Volume -- (6) Astrophysical Gas Dynamics: Shocks 6/47
For a perfect gas there is an adiabatic relationship between pressure and specific volume, viz. p constant (7) Similarly, there is a relationship between P and for shocks - the shock adiabat. p p p Astrophysical Gas Dynamics: Shocks 7/47
The purpose of the following is to derive the shock adiabat. Continuity of mass flux implies v x j ----- j v x j (8) Using these expressions we can write the xx component of the momentum flux as: p v + x p + --j p + j (9) Astrophysical Gas Dynamics: Shocks 8/47
Substitute these expressions into the momentum flux equation allows us to solve for j: p + j p + j j p p ----------------- p ------ (0) i.e. the mass flux through the shock is determined by the difference in pressures and specific volumes. Velocity difference We have for the velocities on either side of the shock: v x j v x j () Astrophysical Gas Dynamics: Shocks 9/47
Hence the velocity difference is given by: v x v x j and we can write the mass flux as j v x v x ---------------------------------- p p ----------------- () (3) Hence the velocity difference is given by: v x v x p p p p p (4) Astrophysical Gas Dynamics: Shocks 0/47
Thus, the velocity difference is also determined by the pressure difference and the specific volume difference. Energy equation: Enthalpy difference The difference in enthalpy between the pre-shock and postshock fluids can be determined from the energy equation. First note that the combination of enthalpy and velocity resulting from the continuity equations can be written in terms of the mass flux and specific volume as: h + --v x h + --j (5) Astrophysical Gas Dynamics: Shocks /47
Therefore, the energy continuity equation becomes: h --j + h + --j h h --j (6) Using the expression for developed above in terms of the pressure and specific volume jumps, viz. j j p p ----------------- (7) Astrophysical Gas Dynamics: Shocks /47
we have h h -- p p + Summary of Rankine-Hugoniot relations v x v x j p p ----------------- p p h h -- p p + (8) (9) Astrophysical Gas Dynamics: Shocks 3/47
Note one feature of shocks that is apparent from the above. We require the sign of to be the same as that of p p for the Rankine-Hugoniot relations to be physically valid. Hence if the pressure increases ( p p ) then so also should the density ( ). Also the specific enthalpy increases. For a -law equation of state, h + p ----------- ---------- p -- ---------- kt --------- m (30) Therefore, an increase in enthalpy implies an increase in temperature across a shock. Astrophysical Gas Dynamics: Shocks 4/47
.6 Velocity difference The result v x v x p p (3) whilst derived in the shock frame is actually independent of the frame of the shock. This frame-independent relationship is very useful in a number of contexts. Astrophysical Gas Dynamics: Shocks 5/47
3 The shock adiabat for a polytropic gas In order to complete the relationship between p and we require an equation of state. This provides a relationship between enthalpy and pressure. Since h + p ----------- ---------- p -- ----------p Astrophysical Gas Dynamics: Shocks 6/47 (3) for a polytropic gas, then the last of the Rankine-Hugoniot equations implies that ---------- p p ---- v x v x ----- --------- -- p p + + p + p --------------------------------------------------- p + + p (33)
This equation relating and p is the shock adiabat we have been aiming for. As one can easily see, once p, p and are known then is determined. The inverse of the above relationship is: p ----- p + -------------------------------------------------- + (34) and this equation tells us that if and p are known then p is also determined. Astrophysical Gas Dynamics: Shocks 7/47
3. Temperature For an ideal gas: p kt --------- T m m ------- p k -- (35) Hence, T ----- T p p --------------- ----- ---- p p p ----- + p + p --------------------------------------------------- p p + + p (36) Astrophysical Gas Dynamics: Shocks 8/47
3. Pre- and post-shock velocities Use j p p ----------------- ---- p p ----------------------- p + + p --------------------------------------------------- (37) Astrophysical Gas Dynamics: Shocks 9/47
To determine the velocity v x we use j j p + + p --------------------------------------------------- v x c s --------- + p + ----- p (38) Similarly (just swap and ) v x c s --------- + p + ----- p (39) Astrophysical Gas Dynamics: Shocks 30/47
The pre- and post-shock Mach numbers are given by: M x M x It is easy to show that for + p + ----- p --------------------------------------------- + p + ----- p --------------------------------------------- p p M x and M x (40) (4) Astrophysical Gas Dynamics: Shocks 3/47
i.e. the normal component of pre-shock velocity is supersonic and the normal component of post-shock velocity is subsonic. This reflects the dissipation which occurs at a shock discontinuity which increases the temperature at the expense of the velocity. It is important to recognize that these constraints do not apply to the transverse component of shock velocity since this is arbitrary and conserved. 3.3 Velocity difference We can express the velocity change across a shock in terms of the pressure difference. Astrophysical Gas Dynamics: Shocks 3/47
Previously we had v x v x p p p p / / / ---- and as we have seen, we have for a polytropic gas: ---- + p + p --------------------------------------------------- p + + p (4) (43) Therefore, ---- p p --------------------------------------------------- p + + p (44) Astrophysical Gas Dynamics: Shocks 33/47
and / p p ---------------------------------------------------------------- p + + p / v x v x (45) As before this result is independent of the shock frame. Astrophysical Gas Dynamics: Shocks 34/47
3.4 Shock angles Velocity discontinuities in the frame of the shock y v x v Shock Astrophysical Gas Dynamics: Shocks 35/47
From the above diagram tan v y --------- v x v y --------- tan v x tan -------------- tan Now it is readily shown from the above that v x --------- v x v y --------- v x v y --------- v x (46) v x v x --------- (47) i.e. the fluid velocity bends away from the normal as shown in the diagram. Astrophysical Gas Dynamics: Shocks 36/47
4 Strong shocks The strength of a shock is characterised by the pressure ratio p p. As this ratio becomes infinite one can see from the above expression for that ----- ----------- + (48) and that this is finite. For a monatomic gas this ratio is 4. This limiting case is often used in astrophysics since shocks are often quite strong. Astrophysical Gas Dynamics: Shocks 37/47
Note also the corresponding limit for the velocities: v x --------- v x + ----------- 4 for 5 -- 3 (49) 5 Weak shock waves The properties of weak shocks, as well as being interesting in themselves, can be used to derive interesting properties that are valid for shock waves in general. The following is valid for any equation of state. We take the thermodynamic variable specific enthalpy to be a function of state of the specific entropy and the pressure, i.e. h hp s Astrophysical Gas Dynamics: Shocks 38/47 (50)
Consider first the enthalpy jump in a shock. We expand to first order in the entropy and up to third order in the pressure. (Second order and higher terms in the entropy turn out to be unimportant.) h h h s p s s + h p (5) Now use the thermodynamic relation between enthalpy, entropy and pressure s p p -- h -------- p p p + + -- 3 h 6 -------- p 3 s dh ktds + dp s p p 3 (5) Astrophysical Gas Dynamics: Shocks 39/47
This implies h s h -------- p p s kt h p 3 h -------- p s p 3 and therefore the enthalpy jump is given by: h h kt s s + p p s s -------- p s (53) -- p p p + + -- s 6 -------- p s p p 3 (54) Astrophysical Gas Dynamics: Shocks 40/47
We similarly take the specific volume to be a function of and expand: ps p s p p + -- -------- p s p p + Os s + Os s p p ps (55) The last terms turn out to be unimportant. We now substitute into the relationship between enthalpy and pressure jumps, viz h h -- p p + (56) Astrophysical Gas Dynamics: Shocks 4/47
Writing + + (57) we can put the Rankine-Hugoniot relation for the enthalpy in the form We then put h h p + --p p + -- p p p (58) (59) Astrophysical Gas Dynamics: Shocks 4/47
so that h h p 3 + -- p + -- p p 4 p (60) Equating the expression for the enthalpy jump in terms of the entropy jump and the pressure jump to the expression immediately above obtained from the Rankine-Hogoniot equations, we obtain: kts s p -- p p + + + -- 6 pp p 3 p -- p p + + -- 4 pp p 3 (6) Astrophysical Gas Dynamics: Shocks 43/47
Terms in p cancel out up until the third power, and this is why we need to keep this many terms in p but not in s. The result is Normally the quantity e.g. for p s s K -------- p ----------- kt -------- p -------- p s s 0 Astrophysical Gas Dynamics: Shocks 44/47 s + ----------- p p p 3 (6) (63) (64)
Thus the entropy only increases at a weak shock if p p 0 (65) The second law of thermodynamics tells us that entropy always increases so that for a weak shock p p (66) This relationship holds for a shock of arbitrary strength. However, the proof is rather involved. (See Landau & Lifshitz, Fluid Mechanics.) Astrophysical Gas Dynamics: Shocks 45/47
5. Velocity of a weak shock The mass flux is given by The derivative p j s p ------ p ----------------- p p s ----------- ----- p p --------- p s ----------- c s (67) (68) Therefore j v x v x c s (69) Astrophysical Gas Dynamics: Shocks 46/47
and to first order v x v x (70) i.e. the velocities of the pre- and post-shock gas are equal to the sound speed, or in other words the shock travels at the sound speed with respect to the gas on either side of the shock. These limits are evident for the polytropic case discussed above. c s Astrophysical Gas Dynamics: Shocks 47/47