In two recent publications Polihronov and Straatman   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  . Presently, the literature contains no widely accepted explanation of the temperature separation phenomenon as noted in a recent review by Thakare et al.  , 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  , and later textbooks have offered additional perspectives    . More recent work has focused on two and three-dimensional flow within rotating passages. Tatro and Mollo-Christensen  have studied the Ekman layers at low Rossby number flows experimentally, noting the presence of type I and type II instabilities. Kristofferson and Andersson  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  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  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   . Bosman  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.   .
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  by applying the Galilean transformation technique to the stationary conservation equations as suggested by Kageyama and Hyodo  . Here we work only with the steady forms of these equations. The conservation of mass is
where is the velocity in the rotating and accelerating reference frame, 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, is the angular velocity of the frame (which can be unsteady in general), and is the position vector. 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 is the specific internal energy. Notice only the velocity vector and the position vector 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.
1.2. Auxillary Equations
All fluids analyzed in this work are characterized by the ideal gas equation of state:
where 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 is the volumetric heat capacity and is the isobaric heat capacity. Fluids which obey the ideal gas law and have constant heat capacities are called perfect gases  .
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:
Assuming the fluid is a perfect gas, the mass, momentum, and energy equations become
where the relevant dimensionless groups are defined in Table 1.
Using the same scaled variables the ideal gas Equation (4) becomes
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: ,
3) steady flow,
4) subsonic flow,
5) unidirectional flow along the -axis such that ,
7) adiabatic, and
8) negligible heat conduction.
Based on these assumptions we have neglected any influences listed by Lyman  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
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.
Use of the ideal gas law allows Equations (11) and (13) 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 14 and 16 to obtain the differential equation
whose solution is
Equation (18) is an expression of Bernoulli’s theorem in a rotating framework. Inserting the boundary conditions , and yields
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:
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
Re-dimensionalizing Equation (22) and evaluating at yields the temperature distribution found by Polihronov and Straatman  :
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 Ó software  to demonstrate the accuracy of Equations (15) and (18) over Equation (22). A 1D mesh was generated for a straight square duct containing 103 evenly spaced grid points. Air was chosen as the working fluid, with a heat capacity ratio , 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 (18) and the CFD velocity profile, and a maximum error of was observed between Equation (15) and the CFD temperature profile.
2.2. Arbitrary Cross-Sectional Area
We will now generalize the above results to a duct of varying cross section . Analyzing a thin slice of a straight duct aligned with the axis where the free-slip boundary condition is applied at the duct walls leads to the following governing equations:
Figure 2. Plots of non-dimensional velocity and temperature in a straight duct with and .
Invoking the ideal gas Equation (10), introducing the scaled cross-sectional area , and non-dimensionalizing 24 - 26 yields:
Equations (27) and (29) may be solved by direct integration, and Equation (10) may be used to obtain an expression for the pressure distribution:
Solving Equation (28) requires the use of Equations (30) and (32) to obtain
Equation (33) may be solved using the method of exact differentials, yielding
Three boundary conditions are required to evaluate constants C, D and E. If the duct area is constant (i.e. ) then and Equations (30), (31), (32) and (34) reduce to the solutions for a constant cross-section duct; Equations (14), (15), (16) and (18) respectively. In addition, we note that Equation (34) is in complete agreement with Equation (9) in Ref.  .
To confirm this result, we have conducted several CFD simulations and compared the computed profiles to Equations (31) and (34). 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 103 grid points was generated. The solution was again computed using ANSYS CFX Ó  . 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 (34) and the CFD results while a maximum error of 0.004% was observed between Equation (31) and the CFD results.
3. Rotating Passage
In this section we will further generalize the above results to an arbitrarily curved passage defined by the parameterization
and the scaled path vector is given by . The components of may be any well-behaved functions, producing, for example, the path shown in Figure 5.
The following derivation requires that the axis of rotation contains the origin of the co-ordinate system on which is defined. The unit tangent vector parallel to the path 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:
Figure 3. Schematic of the varying cross-section duct, rotating with a constant angular velocity about the origin O. The area as a function of the co-ordinate is given by Equation (35).
Figure 4. Plots of non-dimensional velocity and temperature of a duct with cross-sectional area varying in accordance with 36 with and .
Figure 5. An arbitrary path defined by , rotating about .
3.1. Constant Cross-Section
The following steps apply when the duct cross-sectional area is constant along the path. If we have some quantity , its total derivative is
Furthermore, given that all quantities are defined only on the path , any gradient (e.g. ) is parallel to the unit tangent vector:
for some unknown value c. We will choose to satisfy Equation (39). Substituting Equations (38) and (40) into the governing Equations (7) and (9) yields (after some manipulation)
Solving 41 and 42 yields Equations (14) and (15), respectively. The ideal gas law may be expressed using Equation (16).
To obtain a general solution for the velocity profile we will take the dot product of Equation (43) with . Since and 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
Equation (44) may be solved using direct integration:
3.2. Arbitrary Cross-Sectional Area
For a rotating passage of arbitrarily varying cross-sectional area , and we must include in a manner similar to Section 2. The density, temperature, and pressure profiles are given by Equations (30), (31), and (32), respectively. The velocity profile is given by
3.3. Radius as the Parameter
In light of this result, we are interested to see if we can make any statements about the function . 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 :
Invoking definition 46 reveals whenever is well-behaved over the desired range of a. Under these circumstances, Equation (47) collapses to 34, and we conclude that the flow speed at any point in a constant cross-section rotating passage under isentropic conditions is a function of the radial position only.
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 and transient energy storage respectively. Furthermore we recognize that 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 :
where and Equation (21) has been used. Equation (53) might be rewritten in terms of a duct tip speed , so that where , the work transferred to/from the passage is given by
Equation (55) is the rate form of the angular rocket propulsion equation developed by Polihronov and Straatman  .
In a curved passage we must express the velocity in the stationary frame as . Substituting this expression into Equation (52) gives
This equation cannot be reduced any further without knowing the form of .
5. 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.
5.1. 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  . In this configuration the Mach number of the flow through the shroud, , should be less than 1:
5.2. 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  :
If the fluid is assumed to be a perfect gas, the total temperature is found by invoking Equation (6)
Equation 60 is similar to Equation (17-4) in the thermodynamics text by Cengel and Boles  , with the inclusion of the rotational energy per unit mass. This quantity is useful in stationary flows because it is constant over isentropic processes.
In contrast, the total temperature in the rotating frame may be defined as:
To see if either of these parameters are constant in the rotating duct problem we insert the temperature profile (Equation (31)) into the velocity profile (Equation (47)) and re-dimensionalize:
Rearranging the above equation gives
We have called the stagnation temperature, as this is the temperature which is attained if the fluid is brought to rest isentropically (while exchanging some energy with the walls of the passage). We have also replaced the parameter a with the co-ordinate , to emphasize that this quantity is the radial distance from the axis of rotation. Upon comparing Equations (59)-(61) it’s clear that neither the total temperature in the stationary frame nor the total temperature in the relative frame are constant along the passage, while the stagnation temperature, , is. Readers familiar with turbomachinery analysis will recognize the quantity as the rothalpy  . Notice that the total temperature and stagnation temperature are not equal in general; i.e. .
The isentropic gas equations may be used to find relationships between stagnation and static pressure and density:
Furthermore, we can use Equation (61) to define the ratio of stagnation to static temperature in terms of dimensionless numbers:
where is a local Mach number and is the tip Mach number. Equation 64 reduces to its stationary counterpart (Equations (16)-(17) in Ref.  ) when .
In addition, we evaluate Equation (47) at the location to devise two useful relationships between Ma, Ro, , and :
5.3. Choked Flow
Using the above definitions, the choked flow condition may be identified. The mass flow rate at any location in a radial passage is given by
Using property ratios 65 and 63 and simplifying yields
We can also define the maximum possible mass flow rate for any given duct, by differentiating Equation (68) with respect to and setting the result equal to 0, which yields . Inserting this restriction into Equation (68) yields the critical, or choked mass flow:
We can nondimensionalize with :
Using Equations (63) and (64), the property ratios in the above equation may be cast in terms of the global Mach number and tip Mach number:
Equation (71) represents the maximum possible mass flow rate at any radial location. Notice varies with the radial co-ordinate . If, at any location, , the flow will be choked in the passage.
While expression 71 is useful, we desire a simpler test to determine whether the flow is choked. Regardless of the area profile the critical mass flow rate in the passage is dictated by the location of minimum choked flow, that is, where Equation (71) is minimized. We begin by differentiating with respect to :
where we have defined the parameter B for compactness:
Setting Equation (72) equal to 0 yields
5.3.1. Constant Cross-Section Duct
For the case when , the above equation suggests the only extrema is at . By differentiating 73 again and inserting gives
Since each of the terms in the above equation are positive, the concavity of 71 is positive at , confirming that is a minima. Since it is the only extrema, it must be the global minimum, and therefore the location which determines the minimum choked mass flow rate for the duct of constant cross-section. Inserting into Equation (71) yields
Therefore, in order to ensure the flow is not choked, we require
Combining Equations (75) and (65) and re-arranging results in a cumbersome inequality in terms of Ma and Ro, which has been plotted in Figure 6.
5.3.2. Rotating Slice
If the duct area varies with the equation , (a rotating slice), Equation (73) reduces to . Since there are no real solutions, there are no extrema on Equation (71), and the critical section for choked flow may be determined by comparing the choked mass flow rates at the inner and outer radii: where . Clearly, , and the maximum mass flow rate is
5.3.3. Critical Duct
If Equation (71) is evaluated such that the flow is choked everywhere ( ), we can formulate the critical cross-sectional area profile:
where we have recognized the appearance of the area ratio , which has been defined for stationary ducts:
We have also introduced a modified tip Mach number,
Equation (78) has been tabulated for many values of Ma and in many engineering texts such as Ref.  . The scale of the profile is determined by , while the profile shape is determined by , which has been plotted in Figure 7. When , and the curves represent the minimum area required to avoid the choked flow condition. When , these curves are scaled by . Notice the required area decreases with increasing . We now propose an alternative method to determine whether the flow is choked in a known duct: plot the profiles and on the same axes. If at any point, the flow will be choked.
In this work we have developed expressions for density, temperature, pressure, and velocity profiles within arbitrarily curved ducts with arbitrarily varying cross-sectional area profiles under isentropic, compressible flow conditions where the fluid may be considered a perfect gas. These profiles are given by Equations (30), (31), (32), and (34) respectively. We have verified our results through comparison with equivalent CFD simulations. These derivations verify the assumption that is frequently made in turbomachinery texts: rothalpy is conserved along curved passages when the five requirements indicated by Lyman
Figure 6. Restrictions on Rossby and Mach numbers dictated by the choked flow constraint for a straight, constant cross-section duct which crosses (or terminates at) the axis of rotation. All equations were evaluated with .
Figure 7. Plots of Equation (78) for a range of (tip Mach numbers) with .
et al.  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, , and 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.
The authors gratefully acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).
A Duct or passage cross-sectional area
a Independent parameter
Critical cross-sectional area profile
Area to throat area ratio in a stationary passage
C, D, E Constants of integration
cp Isobaric heat capacity
cv Volumetric heat capacity
Transient energy storage in a control volume
Unit vectors aligned with the x; y; and z axes, respectively
Mass ow rate
Maximum (choked) mass ow rate
Minimum choked mass ow rate; occurs at the location in a duct or passage which will choke first if the mass ow rate is slowly increased.
p Thermodynamic pressure
P* Parameterized position vector
R8 Specific ideal gas constant
t Unit tangent vector to a parametric curve P*
Flow speed along along a constrained path
x Position vector
X, Y, Z Components of position vector
x, y, z Cartesian co-ordinates
Ratio of speci_c heats
speci_c internal energy
Arbitrary scalar or vector quantity
Angular velocity of rotating frame
Global Mach number
Shroud Mach number
Local Mach number
Tip Mach number
Modified tip Mach number ratio
* Non-dimensional quantity
1 Quantity at boundary nearest to the center of rotation
2 Quantity at boundary furthest from the center of rotation
c Characteristic dimension
in Quantity entering a control volume
out Quantity exiting a control volume
' Derivative of single-variable function
^ Quantity in non-stationary frame
CFD Computational Fluid Dynamics
RHVT Ranque-Hilsch Vortex Tube
RHS Right Hand Side