Compressible flows: linear theory
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Compressible flows: linear theory¶
While most of this course foused on incompressible flows, it is worth spending a little time on compressible flows one encounters in particular in gas dynamics. In this section we’ll focus on the inviscid limit. The Euler equations are as before:
but the continuity equation is
and the final term does not in general vanish. Finally, to close these equations, we will need some kinf of equation of state \(p = p(\rho)\). In general \(p\) will depend on othe themrodynamic variables such as temperature as well, in which case we will need equations for those quantities; we are assuming here that those are fixed. Two candidate equations of state are:
for an isothermal ideal gas; suc gases will have heat exchange between warmer compressed regions and colder rarified regions. Another, for cases of no heat exchange, is the adiabatic gas law
which turns out to be more relevant for sound waves. For this case, we can define the speed of sound \(c_s\) via:
For the equation of state above,
We will see below why we idenify this with the speed of sound waves.
Sound waves¶
Once again we start from a static solution with constant density and pressure: \({\vec v} = 0\), \(p= p_0\), \(\rho = \rho_0\), with backgound speed of sound \( c_{s0}^2 = \gamma p_0/\rho_0\). We expand
The Euler nd mass conservation equations are satisfied to order \({\cal O}(\eps^0)\). At \({\cal O}(\eps)\), the equations are:
Taking thecurl of the first equation, we have \(\partial_t {\vec \nabla} \times \vv_1 = 0\), so hat vorticity is conserved. If we perturb our system with irrotational flow, it remains irrotational, so let us do hat. we can then wrote \(\vv_1 = {\vec\nabla}\phi_1\), and our equations become:
The first equation is in fact a pure gradient:
The term in parentheses is therefore a funciton of time only. defining this as \(D(t)\), if we redfine \(\phi_1 \to \phi_1 - \int_0^t dt' D(t')\), then the velocities do not change, and we have
Taking the time derivative of this equation and inserting the conservation equation we are left with the wave equation
Plane wave solutions¶
Let us consider waves propaating in three dimensions, with boundaries at infinity. We can attempt a separation of variables:
Inserting this into the wave equation and dividing by \(\phi_1\), we find:
Each term depends on an independent variable and thus must be equal to a constant. Thus, we define
and similarly for \(Y,Z\). For the solution to remain finie at infinity, we must have \(k_x^2 > 0\), so that a general solution to the above is
More generally, we have
(the minus sign is conventional). Finally, we have
Defining \(\omega = c_{s0} \sqrt{\vec k}^2\), we have the solution:
These describe longitudinal waves: \({\vec v}_1 = {\vec \nabla} \phi_1\) points in the direction \({\vec k}\).
Note this is a complete set of solutions; in other words, for any initial condition \(\phi_1({\vec x},t=0)\), we can always decompose this via a Fourier integral into plane waves:
from which the time evolution is:
Now let us consider the case of waves moving purely in the \(x\) direction (\(k_y = k_z = 0\)). In this case, \(\omega = c_{s0}|k|\), \(k = \pm |k|\), so the solutions are:
A general solution will be a linear combination of these; we thus find that the general solution as the form:
Another way to see this is to define \(x_{\pm} = x \pm c_{s0} t\). Is is sraightforward to show, using the chain rule, that
From this equation the above general solution follows immediately.
The upshot is that a given initial configuration always splits into a left- and right-moving wave packet, each of which travels at speed \(c_{s0}\) without distortion.
Finally, we can ask when the linnear approximaion is good. In the present case we will ask when \(\eps \rho_1 \ll \rho_0\). Consider a solution at late times such that the left- and right-moving wavepackets are well-=separated, and focus on one such (we will choose \(f_-\) for specificity). Starting with
and using \(\partial_t v_1 = c_{s0} \partial_x v_1\) so that
We set the constant to zero by demanding that the velocity perturbation vanishes when the pressure does (alternatively, by choosing the correct frame of reference). Finally, setting \(p_1 = c_{s0}^2 \rho_1\) we have
From this we see that
In other words the small amoplitude limit corresponds to velocity perturbations much slower than the sound speed.
Spherical waves¶
Returning to three dimensions, let us consider spherically symmetric waves emanating from the origin. This means that \(\phi_1(r,\theta,\phi) = \phi_1(r)\), so that
The wave equation is then
If we set \(phi_1 = f(r)/r\), then
so that a general solution is
which is a sum of waves outgoing from and ingoing to the origin.
Supersonic flows past an airfoil¶
The next steps is to consider a steady (constant-velocity) flow \({\vec v} = U {\hat x}\), \(p_0\), \(\rho = \rho_0\), in the presence of a very thin airfoil. We expect that if the aifoil is sufficiently thin it will induce only a very small perturbation of the basic state, so that a linear theory, first order in perturbations, applies.
In this case we set
We will assume that the airfoil is straight and long enough in the \(y\) direction that the perurbations depend on \(x,z\) only, and that \(v_y \equiv V\) vanishes, leaving us with 2d flow. Setting \(\vv_1 = u_1 {\hat x} + w_1 {\hat z}\), and following the same logic as before, we have the Euler equations at order \({\cal O}(\eps)\):
and the mass conservation equation:
Next, we take the \(x\) derivative of the second of te two Euler equations above, and subtract the \(y\) derivative of the first, to find:
In other words the voriticity is \(x\)-independent at order \({\cal O}(\eps)\). Since it vanishes at \(x \to - \infty\), where the velocity is just \(U{\hat x}\), it vanishes everywhere, and we can consider potential flow \({\vec v}_1 = {\vec \nabla} \phi_1\). Using this in the mass conservation equations we find
where the Laplacian is taken to be two-dimensional. Now once again we take the linearized version of the equation of state: \(p_1 = c_{s0}^2 \rho_1\), plug this into the equation above, and hen use the \(x\) component of the Euler equation to eliminate \(p_1\), and we find
We define the Mach number as \(M = \frac{U}{c_{s0}}\). we cna see that qualitatively the equations loof very different for \(M > 1\) as fo \(M < 1\), as it changes the relative sign of the derivatives.
The next step is to impose boundary conditions. Let the airfoil shape be \(y(x) = \eps f(x)\), with \(y'(x) = \eps f'(x)\) (\(y\) could get large if the wing is too wide, even if \(y'\) is small; we will not consider that possibility). If we demand that the normal velocity vanishes/the velocity is tangent ot the wing, then the boundary condition is
which at leadig order in \(\eps\) becomes
To leading oder in \(\eps\) we can evaluate \(v_1\) at \(y = 0\), so that
Now let us solve this equation for supersonic flow \(M > 1\). In this case, we can define \(\tau = \frac{x}{\sqrt{M^2 - 1}}\) and the equation of motion for \(\phi_1\) becomes
This is essentially identical to the wave equation for sound waves, in which “time” is a rescaled \(x\). We know the general solutions:
which we can rewrite as
where \(\beta = \sqrt{M^2 - 1}\).
In such supersonic flow we assume that the airfoil will generate compression waves which travel at the speed \(c_s\). By causality, we expect the disturbance to trail the leading edge of the airfoil. A basic argument is that at a point in time, the tip of the airfoil moving through the fluid would generate a circular compression wave travelling at speed \(c_s\). But the airfoil is moving faster than this compression wave, so it always leave the wave behind.
We take \(\phi\) to vanish upstream of the airfoil. Now if we take the tip of the airfoil to lie at the origin, we expect \(\phi\) to start changing first at the tip. By causality only \(f_-\) can start to change above the wing (because \(f_+\) nonzero near the tip would imply it is nonvanishing ahead of the airfoil, out to arbitrarily large values of \(-x\) and \(y\)), while only \(f_+\) can be nonzero below it. Thus \(\phi\) will vanish all the way along \(x = \beta y\) for \(y > 0\), and along \(x = - \beta y\) for \(y < 0\). By continuity \(F_+(0) = 0\) above the wing and \(F_-(0) = 0\) below.
Now let the wing have extent \(L\). Imposing the boundary conditions we have \(\phi'_1 = f'_-(x) = - \frac{U}{B} f'(x)\), so that we get \(\phi_1 = f_-(x) = - \frac{U}{\beta} f(x) + c\) for \(y > 0^{+}\), where \(c\) is a constant which is set to zero by demanding \(f_+(0) = 0\). Similarly, below the wing we have \(\phi_1 = f_-(x) = - \frac{U}{\beta} f(x)\). Finally, For \(x > L\) the airfoil vanishes again, so that \(f_{\pm}(x > L) = 0\).
The full solution becomes
This means that the streamlines have the same shape as the airfoil; in particular, \(v_y = \partial_y \phi = U f'(x - \beta y)\) for \(y > 0\), so that \(v_y/U \sim f'\).