Abstract
An experimental investigation of the flow inside a rectangular cylinder with air injected continuously along the wall is performed. This kind of flow is a two-dimensional approximation of what happens inside a solid rocket motor, where the lateral grain burns expelling exhaust gas or in processes with air filtration or devices to attain uniform flows. We propose a brief derivation of some analytical solutions and a comparison between these solutions and experimental data, which are obtained using the particle image velocimetry technique, to provide a global reconstruction of the flowfield. The flow, which enters orthogonal to the injecting wall, turns suddenly its direction being pushed towards the exit of the chamber. Under the incompressible and inviscid flow hypothesis, two analytical solutions are reported and compared. The first one, known as Hart–McClure solution, is irrotational and the injection velocity is non-perpendicular to the injecting wall. The other one, due to Taylor and Culick, has non-zero vorticity and constant, vertical injection velocity. The comparison with laminar solutions is useful to assess whether transition to turbulence is reached and how the disturbance thrown in by the porous injection influences and modifies those solutions.
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Notes
We proved that this form of \(\psi\) is a consequence of the conservation of mass under the zero vorticity hypothesis. Without this hypothesis, we are not able to prove that \(\beta\) is zero.
In addition, the error should be scaled by \(U_\mathrm {inj}\). Then, we obtain a maximum error, the one relative to the third section, of the order of 0.06.
Abbreviations
- A :
-
Area of the porous plate; \(A = {120}\hbox { cm}^2\)
- \(A_0\) :
-
Amplitude of acoustic waves
- \(B_j\) :
-
jth bin of the histogram representation
- H :
-
Height of the channel; \(H = {2}\hbox { cm}\)
- I :
-
Turbulence intensity; \(I = \sqrt{\langle u'u'\rangle + \langle v'v'\rangle }\)
- \(I_{\mathrm {p}}\) :
-
Turbulence intensity peak; \(\max _y I\)
- L :
-
Length of the channel; \(L = {24}\hbox { cm}\)
- \(N_\mathrm {bin}\) :
-
Total number of bins
- P :
-
Divergence of a two-dimensional flow, first invariant of \(\nabla \mathbf {u}\); \(P = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\)
- Q :
-
Third invariant of \(\nabla \mathbf {u}\); \(Q = \frac{\partial u}{\partial x}\frac{\partial v}{\partial y} - \frac{\partial u}{\partial y}\frac{\partial v}{\partial x}\)
- \(U_\mathrm {inj}\) :
-
Injection velocity
- \({\varDelta }\) :
-
Discriminant of the eigenvalues characteristic equation for the two-dimensional \(\nabla \mathbf {u}\); \({\varDelta }= P^2 - 4Q\)
- \(\varOmega _z\) :
-
Transversal component of vorticity
- \(\bar{\cdot }\) :
-
Spatial average; i.e., a discrete approximation of the integral \(\frac{1}{L}\int _0^L s(x) \,\mathrm{{d}}x\)
- \(\cdot '\) :
-
Fluctuation of a variable with respect the mean
- \(\cdot ^*\) :
-
Dimensional variable
- \(\dot{m}\) :
-
Volume flow rate
- \(\lambda\) :
-
Wavelength
- \(\langle \cdot \rangle\) :
-
Ensemble average; e.g., \(\langle s \rangle = \frac{1}{N} \sum \nolimits _{i=1}^N s_i\) where N is the number of data
- \(Re_{\mathrm {inj}}\) :
-
Injection Reynolds number; \(Re_{\mathrm {inj}} = HU_{\mathrm {inj}}/\nu\)
- \(\nu\) :
-
Kinematic viscosity of fluid
- \(\psi\) :
-
Stream function
- \({\varvec{\Omega }}\) :
-
Vorticity
- \(\mathbf {u}\) :
-
Velocity vector
- \(h_i\) :
-
ith horizontal acoustic eigenmode; \(h_i =A_0 \sin \left( i\frac{2\pi }{4L}x\right)\)
- k :
-
Wavenumber
- p :
-
Pressure
- u :
-
Horizontal component of velocity
- v :
-
Vertical component of velocity
- \(v_i\) :
-
ith vertical acoustic eigenmode; \(v_i = A_0\sin \left( i\frac{2\pi }{2H}x\right)\)
- w :
-
Transversal component of velocity
- x :
-
Horizontal direction. Distance with the head end of the channel
- y :
-
Vertical direction
- z :
-
Transversal direction
- HC:
-
Hart–McClure solution
- PDF:
-
Probability density function
- TC:
-
Taylor–Culick solution
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Perrotta, A., Romano, G.P. & Favini, B. An experimental study of wall-injected flows in a rectangular cylinder. Exp Fluids 59, 11 (2018). https://doi.org/10.1007/s00348-017-2472-1
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DOI: https://doi.org/10.1007/s00348-017-2472-1