Compressible and Choked Flows in Rotating Passages ()

The present study revisits the rotating duct problem examined by Polihronov
and Straatman (J. Polihronov and A. G. Straatman, *Phys. Rev. Lett*. v. 109, p.
054504 (2012)). Starting from the general compressible Euler equations in a
non-stationary reference frame closed form expressions for velocity, temperature,
density and pressure along the duct are determined. The present results
are more general than those obtained by Polihronov and Straatman, as the
change of in-frame kinetic energy has been retained. The improvement of
the present results over Polihronov and Straatman’s is demonstrated by
comparison with the results of a computational fluid dynamics study. The
new results have been further generalized to the case of a rotating duct with
varying cross-sectional area, and again for a general curved passage in
three-dimensional space. The work required or derived from the rotating
duct has also been computed. The choked flow condition within the passage
of varying cross-sectional area has been identified, along with the constraints
which must be placed on the Mach, Rossby, and tip Mach numbers to avoid
choked flow. Finally, a straightforward technique to identify any locations
where an ideal rotating flow in a constrained passage will become sonic has
been presented.

KEYWORDS

Cite this paper

Dyck, N. and Straatman, A. (2019) Compressible and Choked Flows in Rotating Passages. *Open Journal of Fluid Dynamics*, **9**, 1-21. doi: 10.4236/ojfd.2019.91001.

1. Introduction

In two recent publications Polihronov and Straatman [1] [2] have applied heuristic techniques to examine the energetics of confined fluid flow in a rotating reference frame. These works were completed in an effort to shed new light on the temperature separation phenomenon within the Ranque-Hilshe Vortex Tube (RHVT), first discovered by Ranque [3] . Presently, the literature contains no widely accepted explanation of the temperature separation phenomenon as noted in a recent review by Thakare et al. [4] , but a fundamental understanding of rotating compressible flows appears to be a promising starting point.

Studies of rotating flows may be divided into two broad categories: flows through rotating passages, and swirling flows. Both types of flows share similar features, but the latter comes with increased complexity. We emphasize that the present work focuses on flows through rotating passages, and will tackle swirling flows in future publications.

Rotating incompressible flows in confined passages have been studied extensively, both analytically and numerically. An initial treatment of rotating flows has been provided by Greenspan [5] , and later textbooks have offered additional perspectives [6] [7] [8] . More recent work has focused on two and three-dimensional flow within rotating passages. Tatro and Mollo-Christensen [9] have studied the Ekman layers at low Rossby number flows experimentally, noting the presence of type I and type II instabilities. Kristofferson and Andersson [10] have employed direct numerical simulations to study turbulent boundary layer flows inside rotating passages, finding the variation in mean velocity profiles with changes in Rossby number. Khesghi and Scriven [11] have used the finite element method to study rotating flows when neither the Ekman nor the Rossby numbers may be neglected, and revealed the presence of an inviscid core flow near the axis of the straight passage.

Outside of the publications by Polihronov and Straatman, rotating compressible flows in confined passages have received attention from a variety of research fields. Most notably, Seymour Lieblein submitted a NACA Technical Note in 1952 [12] wherein he developed a set of equations describing compressible flow in radial compressor blade passages, including a discussion of supersonic flow and the effects of losses. In later publications, it has become popular to define the rothalpy of a compressible fluid undergoing radial motion, wherein the rothalpy has been shown to be constant when the flow may be considered adiabatic and frictionless [13] [14] . Bosman [15] later showed that, for “all engineering intents and purposes”, the error associated with the constant rothalpy assumption may be neglected. Discussions of rothalpy now appear in graduate level fluid mechanics texts such as Refs. [7] [8] .

The objective of the present work is to re-analyze the rotating duct problem studied by Polihronov and Straatman, starting instead from the governing equations of fluid mechanics. We will systematically obtain closed form mathematical expressions for the density, temperature, pressure, and velocity profiles within rotating, one-dimensional, straight and curved passages with constant and spatially varying cross-sectional areas, under the assumption that the flow is compressible, adiabatic, and inviscid. The motivation for this work is to gain insight from the solutions about the mechanism responsible for the temperature separation phenomenon in the RHVT.

1.1. Governing Equations

The conservation equations of mass, momentum, and energy have been appropriately transformed into a general, non-inertial reference frame by Combrinck and Dala [16] by applying the Galilean transformation technique to the stationary conservation equations as suggested by Kageyama and Hyodo [17] . Here we work only with the steady forms of these equations. The conservation of mass is

$\nabla \cdot \left(\rho \stackrel{^}{u}\right)=\mathrm{0,}$ (1)

where $\stackrel{^}{u}$ is the velocity in the rotating and accelerating reference frame, $\rho $ is the density, and $\nabla $ is the gradient operator. The inviscid, steady, conservation of momentum equation in a non-accelerating rotating frame in the absence of body forces is

$\left(\stackrel{^}{u}\cdot \nabla \right)\stackrel{^}{u}=-\frac{\nabla p}{\rho}+\underset{\begin{array}{c}\text{Coriolis}\\ \text{acceleration}\end{array}}{\underset{\ufe38}{2\stackrel{^}{u}\times \Omega}}\underset{\begin{array}{c}\text{centrifugal}\\ \text{acceleration}\end{array}}{\underset{\ufe38}{-\left(\stackrel{^}{x}\times \Omega \right)\Omega}}\mathrm{,}$ (2)

where p is the thermodynamic pressure, $\Omega $ is the angular velocity of the frame (which can be unsteady in general), and $\stackrel{^}{x}$ is the position vector. $\stackrel{^}{x}$ is defined relative to the origin of a co-ordinate system about which rotation occurs. When heat conduction and external heat sources may be neglected, the conservation of internal energy is

$\rho \stackrel{^}{u}\cdot \nabla \epsilon =-p\left(\nabla \cdot \stackrel{^}{u}\right)\mathrm{,}$ (3)

where $\epsilon $ is the specific internal energy. Notice only the velocity vector $\stackrel{^}{u}$ and the position vector $\stackrel{^}{x}$ have been assigned the ${}^{\wedge}$ symbol. This emphasizes that these quantities are transformed versions of their stationary frame counterparts. All other quantities under consideration are scalars, which are not affected by the transformation into the rotating frame, so the distinction between scalar quantities in the rotating frame and their counterparts in the stationary frame is not made.

1.2. Auxillary Equations

All fluids analyzed in this work are characterized by the ideal gas equation of state:

$p=\rho {R}_{s}T,$ (4)

where ${R}_{s}$ is the specific ideal gas constant, and T is the static, absolute temperature.

We will further assume the heat capacities are constant, so that the internal energy and enthalpy may be respectively written as

$\epsilon ={c}_{v}T,$ (5)

$h={c}_{p}T,$ (6)

where ${c}_{v}$ is the volumetric heat capacity and ${c}_{p}$ is the isobaric heat capacity. Fluids which obey the ideal gas law and have constant heat capacities are called perfect gases [18] .

1.3. Nondimensionalization

To further generalize our results we have presented much of our analyses and solutions in terms of non-dimensional quantities. We use the following scaled variables to non-dimensionalize the governing and auxillary equations:

${\stackrel{^}{x}}^{*}=\frac{\stackrel{^}{x}}{{x}_{c}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{u}}^{*}=\frac{\stackrel{^}{u}}{{u}_{c}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}^{*}=\frac{p}{{\rho}_{c}{x}_{c}{u}_{c}{\Omega}_{c}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Omega}^{*}=\frac{\Omega}{{\Omega}_{c}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{*}=\frac{T}{{T}_{c}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\rho}^{*}=\frac{\rho}{{\rho}_{c}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\nabla}^{*}={x}_{c}\nabla $

Assuming the fluid is a perfect gas, the mass, momentum, and energy equations become

${\nabla}^{*}\cdot \left({\rho}^{*}{\stackrel{^}{u}}^{*}\right)=0,$ (7)

$Ro\left({\stackrel{^}{u}}^{\mathrm{*}}\cdot {\nabla}^{\mathrm{*}}\right){\stackrel{^}{u}}^{\mathrm{*}}=-\frac{{\nabla}^{\mathrm{*}}{p}^{\mathrm{*}}}{{\rho}^{\mathrm{*}}}+2{\stackrel{^}{u}}^{\mathrm{*}}\times {\Omega}^{\mathrm{*}}-\frac{1}{Ro}\left({\stackrel{^}{x}}^{\mathrm{*}}\times {\Omega}^{\mathrm{*}}\right)\times {\Omega}^{\mathrm{*}}\mathrm{,}$ (8)

${\rho}^{\mathrm{*}}Ro{\stackrel{^}{u}}^{\mathrm{*}}\cdot {\nabla}^{\mathrm{*}}{T}^{\mathrm{*}}=-\gamma \left(\gamma -1\right)M{a}^{2}{p}^{\mathrm{*}}\left({\nabla}^{\mathrm{*}}\cdot {\stackrel{^}{u}}^{\mathrm{*}}\right)\mathrm{,}$ (9)

where the relevant dimensionless groups are defined in Table 1.

Using the same scaled variables the ideal gas Equation (4) becomes

${p}^{*}=\frac{Ro}{\gamma M{a}^{2}}{\rho}^{*}{T}^{*}.$ (10)

2. Rotating Duct

This section derives the general solution for compressible flow inside a rotating duct under the following assumptions:

1) constant thermophysical properties,

2) steady rotation about the z-axis: $\Omega =\omega \stackrel{^}{k}$,

3) steady flow,

4) subsonic flow,

5) unidirectional flow along the $\stackrel{^}{x}$ -axis such that $\stackrel{^}{u}=\stackrel{^}{u}\stackrel{^}{i}$,

6) inviscid,

7) adiabatic, and

8) negligible heat conduction.

Based on these assumptions we have neglected any influences listed by Lyman [14] which may change the rothalpy inside the duct. A schematic of the duct under consideration is shown in Figure 1.

2.1. Constant Cross-Section

If the cross-sectional area of the duct is constant, the steady, non-dimensional conservation equations of mass, momentum, and energy reduce to

${\stackrel{^}{u}}^{\mathrm{*}}\frac{\text{d}{\rho}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}+{\rho}^{\mathrm{*}}\frac{\text{d}{\stackrel{^}{u}}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}=\mathrm{0,}$ (11)

$Ro{\stackrel{^}{u}}^{\mathrm{*}}\frac{\text{d}{\stackrel{^}{u}}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}=-\frac{1}{{\rho}^{\mathrm{*}}}\frac{\text{d}{p}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}+\frac{{\stackrel{^}{x}}^{\mathrm{*}}}{Ro}\mathrm{,}$ (12)

Figure 1. Schematic of the constant cross-section duct, rotating with a constant angular velocity about the origin O. Here the flow is shown moving from the outer position 2 to the inner position 1, however our analysis is independent of the flow direction. Furthermore, while we have chosen characteristic quantities at position 2, the choice is arbitrary, as long as they are all at the same location.

Table 1. Relevant dimensionless groups.

$Ro\text{\hspace{0.05em}}{\rho}^{\mathrm{*}}{\stackrel{^}{u}}^{\mathrm{*}}\frac{\text{d}{T}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}=-\gamma \left(\gamma -1\right)M{a}^{2}{p}^{\mathrm{*}}\frac{\text{d}{\stackrel{^}{u}}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}\mathrm{.}$ (13)

Use of the ideal gas law allows Equations (11) and (13) to be simplified and solved through direct integration.

${\rho}^{*}=\frac{C}{{\stackrel{^}{u}}^{*}},$ (14)

${T}^{*}=\frac{D}{{\stackrel{^}{u}}^{*\gamma -1}},$ (15)

where C and D are constants of integration. The pressure distribution is therefore given by

${p}^{\mathrm{*}}=\frac{Ro}{\gamma M{a}^{2}}\frac{CD}{{\stackrel{^}{u}}^{\mathrm{*}\gamma}}\mathrm{.}$ (16)

Solving Equation (12) requires substitution of 14 and 16 to obtain the differential equation

${\stackrel{^}{u}}^{\mathrm{*}}\frac{\text{d}{\stackrel{^}{u}}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}=\frac{1}{M{a}^{2}}\frac{D}{{\stackrel{^}{u}}^{\mathrm{*}\gamma}}\frac{\text{d}{\stackrel{^}{u}}^{\mathrm{*}}}{\text{d}{\stackrel{^}{x}}^{\mathrm{*}}}+\frac{{\stackrel{^}{x}}^{\mathrm{*}}}{R{o}^{2}}\mathrm{,}$ (17)

whose solution is

$\frac{{\stackrel{^}{u}}^{\mathrm{*2}}}{2}+\frac{1}{\left(\gamma -1\right)M{a}^{2}}\frac{D}{{\stackrel{^}{u}}^{\mathrm{*}\gamma -1}}-\frac{{\stackrel{^}{x}}^{\mathrm{*2}}}{2R{o}^{2}}=E\mathrm{.}$ (18)

Equation (18) is an expression of Bernoulli’s theorem in a rotating framework. Inserting the boundary conditions ${T}^{\mathrm{*}}\left(-1\right)=1$ , and ${\stackrel{^}{u}}^{\mathrm{*}}\left(-1\right)=1$ yields

${T}^{*}=\frac{1}{{\stackrel{^}{u}}^{*\gamma -1}},$ (19)

$\frac{1}{2}\left({\stackrel{^}{u}}^{*2}-1\right)+\frac{1}{\left(\gamma -1\right)M{a}^{2}}\left(\frac{1}{{\stackrel{^}{u}}^{*\gamma -1}}-1\right)-\frac{1}{2R{o}^{2}}\left({\stackrel{^}{x}}^{*2}-1\right)=0.$ (20)

It is interesting to note that the velocity and temperature profiles are completely independent of the pressure and density. Only the inlet temperature and velocity boundary conditions influence the solution. Equation (20) may also be re-dimensionalized for better understanding of each of the terms:

$\underset{\begin{array}{c}\text{linear}\\ \text{kineticenergy}\end{array}}{\underset{\ufe38}{\frac{{\stackrel{^}{u}}^{2}-{\stackrel{^}{u}}_{c}^{2}}{2}}}+\underset{\text{enthalpy}}{\underset{\ufe38}{{c}_{p}\left(T-{T}_{c}\right)}}-\underset{\begin{array}{c}\text{rotational}\\ \text{kineticenergy}\end{array}}{\underset{\ufe38}{\frac{{\omega}^{2}\left({\stackrel{^}{x}}^{2}-{x}_{c}^{2}\right)}{2}}}=0.$ (21)

When the Mach and Rossby numbers are very small, the linear kinetic energy term in Equation (20) may be neglected and the temperature profile reduces to