Question 2.4: Use of the Continuity Equation (Two Problems: A and B) (A) F......
Use of the Continuity Equation (Two Problems: A and B)
(A) For steady laminar fully-developed pipe flow of an incompressible fluid (see Sect. 2.4), the axial flow is:
\mathrm{{v_{z}(r)=\mathrm{v}_{\mathrm{max}}[1-(\frac{r}{r_{0}})^{2}]}} (E.2.4.1)
Show that the radial (or normal) velocity \rm v_r = 0.
(B) Consider 2-D steady laminar symmetric flow in a smooth converging channel where axial velocity values were measured at five points (see Sketch). Estimate the fluid element acceleration ax at point C as well as the normal velocity v at point B’. All distances are 2 cm and the centerline velocities are 5 m/s at A; 7 m/s at B; 10 m/s at C; and 12m/s at D.
| Assumptions | Sketch |
| • Steady implies \rm\frac{\partial}{\partial t} \equiv 0 |
\underline{\text{Note}} : v_{max} = 2v_{average} |
| • Incompressible fluid: ρ = ¢ | |
| • Axisymmetric pipe: \frac{\partial}{\partial \Theta } \equiv 0 | |
| • Fully developed flow: \rm\frac{\partial}{\partial z} \equiv 0 |
| Concept | Sketch |
| • Approximate via finite differencing the reduced acceleration and continuity equations. |  |
(A)
Based on the assumptions, Eq. (2.10) is appropriate and reads in cylindrical coordinates (App. A):
\rm\frac{\partial \rho}{\partial t} +\nabla\cdot(\rho\vec{\bf v})=0 (2.10)
\rm\frac{1}{r} \frac{\partial(r\mathrm{v}_r)}{\partial r} +\frac{1}{r} \frac{\partial\mathrm{v}_\Theta }{\partial \Theta } +\frac{\partial\mathrm{v}_z}{\partial z} =0 (E.2.4.2)
Clearly, the given velocity profile \rm v_z (r) (see Eq. (E.2.4.1)) is not a function of z, i.e., \rm ∂v_z/∂z =0 which implies with ∂/∂Θ = 0 (axisymmetry) that Eq. (E.2.4.2) reduces to:
\rm\frac{\partial(r{v}_r)}{\partial r} =0Partial integration yields:
\rm{v}_r=f(z)/rwhere \rm 0 ≤ r ≤ r_0 , i.e., r could be zero. That fact and the boundary condition \rm v_r ( r=r_0) = 0 , i.e., no fluid penetrates the pipe wall, forces the physical solution \rm v_r ≡ 0. Indeed, if \rm v_r\neq 0 , such a radial velocity component would alter the axial velocity profile to \rm v_z= v_z(r, z) which implies developing flow; that happens, for example, in the pipe’s entrance region or due to a porous pipe wall through which fluid can escape or is being injected.
(B)
From Sect. 1.3 (Eq. 1.12) the axial acceleration can be written as:
\frac{d^2\vec{\bf r}}{dt^2} =\frac{d\vec{\bf v}}{dt} =\vec{\bf a} (1.12)
\rm a_\mathrm{x}\frac{\partial u}{\partial t} +u\frac{\partial u}{\partial \mathrm{x}} +v\frac{\partial u}{\partial \mathrm{y}} +w\frac{\partial u}{\partial \mathrm{z}} (E.2.4.3)
Based on the stated assumptions \rm (\frac{\partial}{\partial t}=0,\,\frac{\partial}{\partial y}=0,\, w=0) :
a_{x}=u\frac{\partial u}{\partial x}\approx u\frac{\Delta u}{\Delta x}=u_{c}\,\frac{u_{D}-u_{B}}{x_{D}-x_{B}}=10\frac{12-7}{0.04}\\ \rm\therefore \qquad\qquad \boxed{a_x\big|_C\approx 1250\,m/s^2}In order to find \rm v\big|_B, we employ the 2-D continuity equation in rectangular coordinates:
\rm \frac{\partial u}{\partial x} +\frac{\partial \mathrm{v}}{\partial \mathrm{y}} =0 (E.2.4.4)
which can be approximated as:
\rm \frac{\Delta {v}}{\Delta {y}} =-\frac{\Delta u}{\Delta {x}}or
\rm\Delta {v}=\frac{10-5}{0.04} 0.02=-2.5\,m/sRecall that \rm \Delta {v}={v}\Big|_{B^\prime}-{v}\Big|_{B} where \rm{v}_B\equiv 0 <symmetry> so that
\boxed{\rm{v}\Big|_{B^\prime}=-2.5\,m/s}Comment: This is a very simple example of finite differencing where derivatives are approximated by finite differences of all variables. Discretization of the governing equations describing the conservation laws is the underlying principle of CFD (computational fluid dynamics) software (see Sect. 10.2).