How is the energy dissipated by drag split between sound waves and turbulence? Consider an object of radius $R$ traveling at a constant, slightly subsonic or supersonic speed through a homogeneous fluid, in conditions of very high Reynolds number. Because the object experiences a drag force, the power required to maintain its speed is
$$P = F v = \frac12 \rho C_D A v^3.$$
By energy conservation, this energy has to end up in the fluid, and there are clearly at least two places it can go: sound waves that quickly propagate far from the object, originating from the pressure it exerts on the fluid, and heating in its wake, originating from the turbulent motion behind it.
Is there a way to estimate what fraction of the energy goes into each channel, as a function of the Mach number? Or at least some empirical results?
 A: There are some useful reviews/discussions in Ulmschneider [1967], Ulmschneider [1971], Arons and Yennie [1948], and Gross [1965] on the partition of energy in shock waves.  They tend to focus on collisionless shocks in plasmas or ionizing shocks caused by something like a nuclear blast, but they are useful nonetheless (plus you can just drop/ignore the ionized and magnetic field parts for a regular fluid).

By energy conservation, this energy has to end up in the fluid, and there are clearly at least two places it can go: sound waves that quickly propagate far from the object, originating from the pressure it exerts on the fluid, and heating in its wake, originating from the turbulent motion behind it.

Yes, so you can treat the partition of energy across a shock wave in several ways.  Sometimes these details are kind of swept under the metaphorical rug, if you will, in how we choose the form of the enthalpy in the energy flux density conservation relation of the Rankine–Hugoniot relations.  However, Ulmschneider [1967] illustrates that we can consider an energy balance approach that satisfies:
COND = FLOW + RAD - DISS
where COND is the conduction energy flux density, FLOW is the bulk flow kinetic energy flux density, RAD is the radiated energy flux density, and DISS is the dissipated energy flux density (e.g., energy converted to heat).  A slightly more quantitative expression [e.g., see Ulmschneider, 1971] is given by the gradients of the energy flux densities as:
$$
\nabla \left( F_{mech} + F_{cond} - F_{rad} \right) = 0 \tag{0}
$$
where $F_{mech}$ is the mechanical energy flux density that includes both dissipative and flow terms.  Note that the wave pressure generated by absorption and reflection of sound waves [e.g., see Ulmschneider, 1971] is related to $F_{mech}$ given by:
$$
\nabla P_{wave} \approx \frac{ A_{r} }{ C_{s} } \nabla F_{mech} \tag{1}
$$
where $A_{r}$ is a value between 1 (pure dissipation/absorption) and 2 (pure reflection), $C_{s}$ is the speed of sound, and $P_{wave}$ is the wave pressure.  Across a shock wave, Ulmschneider [1971] states that the magnitude of $\nabla P_{wave}$ is only about 1% of the gradient in gas pressure, i.e., it is not a main source in the total energy budget (at least not for shocks in the corona).
Typically the $F_{rad}$ is reserved for waves radiated by the shock like electromagnetic radiation, but in principle, one could rearrange things and treat sound waves as part of this term.  However, as stated above, the discontinuous gas pressure jump at the shock is nearly two orders of magnitude larger than the sound waves.

Is there a way to estimate what fraction of the energy goes into each channel, as a function of the Mach number? Or at least some empirical results?

Kind of, yes.  However, the result is often specific to the problem one examines.  For instance, the magnitude of importance of electromagnetic radiation is very important for supernova shock waves but almost negligible for a shock wave around a fighter jet.  We can calculate the magnitude of the gas pressure jump directly from the Rankine–Hugoniot relations (assuming an appropriate equation of state and polytropic index, of course).  If the ratio of gas to wave pressure gradients is always very large, then one can safely ignore the wave contributions.
In general, the conduction term acts as a sort of "...reservoir from which we can borrow energy to radiate or increase the thermal and kinetic energy..." [Ulmschneider, 1967] of the downstream system.
Note:  I am not being more specific about some of these things because it gets extremely messy very quickly and any generic analytical relationship requires a lot of assumptions and symbol definitions with background information.  In many realistic scenarios, all of this would require empirical measurements.  So I am taking the constructively lazy approach and citing refereed papers below.
References

*

*Arons, A.B., and D.R. Yennie "," Rev. Modern Phys. 20(3), pp. 519--536, doi:10.1103/revmodphys.20.519, 1948.

*Gross, R.A. "Strong Ionizing Shock Waves," Rev. Modern Phys. 37(4), pp. 724--743, doi:10.1103/revmodphys.37.724, 1965.

*Ulmschneider, P. "The Structure of the Outer Atmosphere of the Sun and of Cool Stars," Z. Astrophys. 67, pp. 193--218, url:https://ui.adsabs.harvard.edu/abs/1967ZA.....67..193U/abstract, 1967.

*Ulmschneider, P. "On the Computation of Shock Heated Models for the Solar Chromosphere and Corona," Astron. and Astrophys. 12, pp. 297--309, url:https://ui.adsabs.harvard.edu/abs/1971A&A....12..297U/abstract, 1971.

