Compressible and choked flows in rotating passages Compressible and choked flows in rotating passages 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.


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 Open Journal of Fluid Dynamics (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.  [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.

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 where û is the velocity in the rotating and accelerating reference frame, ρ is the density, and ∇ is the gradient operator. The inviscid, steady, conservation of momentum equation in a non-accelerating rotating frame in the absence of body forces is where p is the thermodynamic pressure, Ω is the angular velocity of the frame (which can be unsteady in general), and x is the position vector. 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 where ε is the specific internal energy. Notice only the velocity vector û and the position vector x have been assigned the ∧ 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.

Auxillary Equations
All fluids analyzed in this work are characterized by the ideal gas equation of state: where s R 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 where v c is the volumetric heat capacity and p c is the isobaric heat capacity. Fluids which obey the ideal gas law and have constant heat capacities are called

1ˆˆ2
, where the relevant dimensionless groups are defined in Table 1.
Using the same scaled variables the ideal gas Equation (4)

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: ω = k Ω , 3) steady flow, 4) subsonic flow, 5) unidirectional flow along the x -axis such that û = u 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.

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    Heat capacity ratio Use of the ideal gas law allows Equations (11) and (14) to be simplified and solved through direct integration. * * , where C and D are constants of integration. The pressure distribution is therefore given by Solving Equation (12) requires substitution of 15 and 17 to obtain the differential equation Equation (19) is an expression of Bernoulli's theorem in a rotating framework. Inserting the boundary conditions ( ) 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 (21) may also be re-dimensionalized for better understanding of each of the terms: When the Mach and Rossby numbers are very small, the linear kinetic energy term in Equation (21) may be neglected and the temperature profile reduces to Re-dimensionalizing Equation (23) and evaluating at ˆ0 x = yields the temperature distribution found by Polihronov and Straatman [1]: This indicates their analysis has implicitly assumed the compressibility of the fluid is small, and the rotational energy of the fluid is large.
We have performed several computational fluid dynamics (CFD) simulations of rotating duct model using ANSYS-CFX  software [19] to demonstrate the accuracy of Equations (16) and (19) over Equation (23). A 1D mesh was generated for a straight square duct containing 10 3 evenly spaced grid points. Air was chosen as the working fluid, with a heat capacity ratio 1.4 γ = , and a free slip boundary condition was enforced at each of the duct walls. The average residuals for the solution were converged within 10 −4 . The results are shown in Figure 2.
A maximum error of 0.03% was observed between Equation (19) and the CFD velocity profile, and a maximum error of was observed between Equation (16) and the CFD temperature profile.

Arbitrary Cross-Sectional Area
We will now generalize the above results to a duct of varying cross section ( ) A x . Analyzing a thin slice x of a straight duct aligned with the x axis where the free-slip boundary condition is applied at the duct walls leads to the following governing equations: Invoking the ideal gas Equation (10), introducing the scaled cross-sectional , and non-dimensionalizing 25 -27 yields: Equations (28) and (30) may be solved by direct integration, and Equation (10) may be used to obtain an expression for the pressure distribution: Solving Equation (29) requires the use of Equations (31) and (33) to obtain  (15), (16), (17) and (19) respectively. In addition, we note that Equation (35) is in complete agreement with Equation (9) in Ref. [12].
To confirm this result, we have conducted several CFD simulations and compared the computed profiles to Equations (32) and (35). These simulations were similar to those described in section 1 unless otherwise noted. The geometry under consideration is the straight square duct depicted in Figure 3 whose cross-sectional area is given by A 1D mesh of constant grid spacing with 10 3 grid points was generated. The solution was again computed using ANSYS CFX  [19]. Solutions were converged when the average residuals were reduced below 10 −4 . The results have been plotted in Figure 4. A maximum error of 0.6% was observed between Equation (35) and the CFD results while a maximum error of 0.004% was observed between Equation (32) and the CFD results.

Rotating Passage
In this section we will further generalize the above results to an arbitrarily curved passage defined by the parameterization The following derivation requires that the axis of rotation contains the origin of the co-ordinate system on which * p is defined. The unit tangent vector parallel to the path p is given by Similarly to the previous derivations, we will neglect the velocity variation across the duct, and assume the velocity at each point is parallel to the unit tangent vector: where * * u = u .

Constant Cross-Section
The following steps apply when the duct cross-sectional area is constant along the path. If we have some quantity Solving 42 and 43 yields Equations (15) and (16), respectively. The ideal gas law may be expressed using Equation (17).
To obtain a general solution for the velocity profile we will take the dot product of Equation (44) with t . Since t and ′ t are orthogonal, the second term on the left hand side must vanish. Furthermore, the first term on the right hand side also evaluates to zero, since it contains a triple scalar product with two parallel vectors. The remaining equation is given by

Arbitrary Cross-Sectional Area
For a rotating passage of arbitrarily varying cross-sectional area ( ) A a , and we must include ( ) * A a in a manner similar to Section 2. The density, temperature, and pressure profiles are given by Equations (31), (32), and (33), respectively. The velocity profile is given by

Radius as the Parameter
In light of this result, we are interested to see if we can make any statements about the function ( ) g a . Consider the following arbitrary curve in a cylindrical co-ordinate system whose z-axis is coincident with the axis of rotation: We will proceed with the parameterization r a = :

Work
One parameter of particular interest is the work derived from a radial turbine (or the work required to drive a radial compressor). In a straight duct, the work is most easily found by writing an energy balance over a control volume enveloping a section of the passage between two points: Because the flow is adiabatic and steady we may neglect the heat transfer Q  and transient energy storage st E  respectively. Furthermore we recognize that 1 2 m m m = =    and insert Equation (6). With these simplifications, we have ( ) ( ) In a straight, radial duct such as the one shown in Figure 1 or the duct shown in Figure 3 we recognize that the velocity in the stationary frame is the vector sum of the in-frame velocity and the local tangential velocity of the duct ( ) This equation cannot be reduced any further without knowing the form of * p .

Choked Flow Limitations
Several assumptions have been employed to arrive at the density, temperature, pressure, and velocity profiles of the above sections. These profiles are therefore only valid for particular combinations of Rossby and Mach numbers. While each of the assumptions listed at the beginning of section 2 merit their own discussion, in this work we will restrict our analysis to the sonic limit. If the flow transitions from subsonic to supersonic at any point in a rotating passage, there will inevitably be a shock at some point downstream as it again becomes subsonic.
Shocks are highly irreversible and therefore undesirable in many applications, therefore it is of great interest to prevent the flow from transitioning in the first place. The next two subsections identify the conditions under which the flow transitions in rotating passages, and develop the appropriate constraints on the selection of Ro and Ma.

Sonic Limitation in the Shroud
Previously the adiabatic duct has been experimentally validated through injecting air tangentially into a circular passage surrounding a rotating disk and allowing the air to expand through radial passages in the disk [20]. In this configuration the Mach number of the flow through the shroud, S Ma , should be less than 1:

Stagnation Properties
In addition, we must ensure the flow does not transition within the passage itself, a state characterized by the presence of choked flow within the passage. To properly define this constraint we must first define several quantities before the topic can be addressed.
First, recall the total enthalpy in the stationary frame is defined as the total energy of a flowing stream per unit mass [21]: Open Journal of Fluid Dynamics  (17)- (18) in Ref. [21]) when 0 ω = .
In addition, we evaluate Equation (48) at the location * 1 r = to devise two useful relationships between Ma, Ro, γ , and We can also define the maximum possible mass flow rate for any given duct, by differentiating Equation (69) We can nondimensionalize with Using Equations (64) where we have defined the parameter B for compactness:

Critical Duct
If Equation (72) where we have recognized the appearance of the area ratio t A A , which has been defined for stationary ducts: We have also introduced a modified tip Mach number, Equation (79) has been tabulated for many values of Ma and γ in many en-Open Journal of Fluid Dynamics gineering texts such as Ref. [21]. The scale of the profile is determined by * t A , while the profile shape is determined by A A < at any point, the flow will be choked.

Conclusion
In this work we have developed expressions for density, temperature, pressure,  Open Journal of Fluid Dynamics et al. [14] are met. In addition, we have characterized the choked flow condition for compressible flow within straight ducts, clearly indicating the constraints on the choice of dimensionless groups Ma, Ro, t Ma , and γ required to avoid the choked flow condition. We have characterized the variation in the critical cross-sectional area and shown how it can be used to quickly evaluate whether or not flow will choke in a rotating duct of known geometry. During this process we have identified the importance of the stagnation temperature, which may be much more pertinent than the often-used total temperature for studies involving rotating compressible flows.