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I am reading "Fundamentals of Aerodynamics" 5th edition, J.D.Anderson. In part 15.6, he said:

Consider a steady two-dimensional, viscous, compressible flow. The x-momentum equation for such a flow is given by Equation (15.19a), which for the present case reduces to: \begin{align} \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = - \frac{\partial p}{\partial x} + \frac{\partial }{\partial y} \left[ \mu \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) \right] \tag{15.27} \end{align}

the equation (15.19a) is: \begin{align} &\rho\frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} + \rho w \frac{\partial u}{\partial z} =\\ &- \frac{\partial p}{\partial x} +\frac{\partial }{\partial x} \left[ \lambda\boldsymbol{\nabla} \cdot\mathbf{V} + 2\mu\frac{\partial u }{\partial x} \right] + \frac{\partial }{\partial y} \left[ \mu \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right) \right] + \frac{\partial }{\partial z} \left[ \mu \left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right) \right] \tag{15.19a} \end{align}

I tried to remove some terms but the second term (indeed, it is $ \partial \tau_{xx}/\partial x$) on the RHS seems not equal to zero for such case. Do you know why do the author ignore this term ?

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  • $\begingroup$ He alludes to "the present case." What exactly is the present case that he is looking at? $\endgroup$ – Chet Miller Aug 5 '18 at 22:01
  • $\begingroup$ @miller the case has been stated: "consider a steady two-dimensional, viscous, compressible flow." If you want to make sure, here you go avionicsengineering.files.wordpress.com/2016/11/… and look at the begining of part 15.6 $\endgroup$ – Dat Aug 6 '18 at 2:01
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    $\begingroup$ It appears that he (tacitly) is neglecting the derivative of $\tau_{xx}$ with respect to x compared to the derivative of $\tau_{xy}$ with respect to y. This is an approximation that is accurate in boundary layer flows such as the one being considered where the main flow is parallel to the wall and the axial velocity varies rapidly with distance away from the wall, but, because of the no-slip boundary condition, varies very gradually (or, at the boundary, not at all) with distance along the wall. $\endgroup$ – Chet Miller Aug 6 '18 at 12:02
  • $\begingroup$ @ChesterMiller That was my thought too when I re-read the chapter. And the equations are the same in the 3rd and 4th éditions, I feel like it would have been caught if it was an error. $\endgroup$ – tpg2114 Aug 6 '18 at 12:08
  • $\begingroup$ No, your thought was definitely correct. This is a standard approach to solving boundary layer problems. But the author should have been more direct and specifically focused on the assumptions and approximations involved. $\endgroup$ – Chet Miller Aug 6 '18 at 12:23
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Following up on @LonelyProf's answer, I think it is not so much a mistake as an oversimplification.

Earlier on pg. 907 after defining the viscous stresses the author goes on to explain:

Once again, the normal stresses are important only where the derivatives $∂_xu$, $∂_yv$, and $∂_zw$ are very large. For most practical flow problems, $τ_{xx}$ , $τ_{yy}$ , and $τ_{zz}$ are small, and hence the uncertainty regarding $λ$ is essentially an academic question. An example where the normal stress is important is inside the internal structure of a shock wave. Recall that, in real life, shock waves have a finite but small thickness. If we consider a normal shock wave across which large changes in velocity occur over a small distance (typically 10−5 cm), then clearly $∂_xu$ will be very large, and $τ_{xx}$ becomes important inside the shock wave.

The section OP is refering to is on dimensional analysis (similarity) and within that context the author is refering to:

flows over two bodies of different shapes...

I think we can safely assume these bodies have a relative large size which together with the quote above results in the assumption that:

$$\tau_{xx} = \lambda\vec{\nabla}\cdot\vec{v}+2\mu\frac{\partial u}{\partial x} \sim 0$$

I think the author blatantly assumed this without proof (e.g. by dimensional analysis) and simplified the expression too quickly. It happens sometimes in technical literature.

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I think that this must be a mistake in the book. The author explicitly states earlier in the chapter that he is taking $\lambda=-\frac{2}{3}\mu$ (Stokes). So one substitutes this into eqn (15.19a) and gets the $x$-component of the standard Navier-Stokes equation, which looks correct to me. And, most important, the second term on the right of eqn (15.19a) does not, in general, vanish!

(Often this equation is written without making the assumption $\lambda=-\frac{2}{3}\mu$, but the mathematical form is made the same by defining $\zeta=\lambda+\frac{2}{3}\mu$ and redefining the pressure $p\rightarrow p-\zeta\mathbf{\nabla}\cdot\mathbf{V}$. So this is not the source of the problem).

I don't see any assumptions other than "steady two-dimensional, viscous, compressible flow", so there seems to be no physical reason for the term in question to be dropped. I guess that it was just omitted by accident. The saving grace is that section 15.6 is just concerned with dimensional arguments.

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