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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.

In two recent publications Polihronov and Straatman [

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 [

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 [

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.

The conservation equations of mass, momentum, and energy have been appropriately transformed into a general, non-inertial reference frame by Combrinck and Dala [

∇ ⋅ ( ρ u ^ ) = 0, (1)

where u ^ 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

( u ^ ⋅ ∇ ) u ^ = − ∇ p ρ + 2 u ^ × Ω ︸ Coriolis acceleration − ( x ^ × Ω ) Ω ︸ centrifugal acceleration , (2)

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

ρ u ^ ⋅ ∇ ε = − p ( ∇ ⋅ u ^ ) , (3)

where ε is the specific internal energy. Notice only the velocity vector u ^ 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.

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

p = ρ 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

ε = 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 [

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:

x ^ * = x ^ x c , u ^ * = u ^ u c , p * = p ρ c x c u c Ω c , Ω * = Ω Ω c , T * = T T c , ρ * = ρ ρ c , ∇ * = x c ∇

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

∇ * ⋅ ( ρ * u ^ * ) = 0 , (7)

R o ( u ^ * ⋅ ∇ * ) u ^ * = − ∇ * p * ρ * + 2 u ^ * × Ω * − 1 R o ( x ^ * × Ω * ) × Ω * , (8)

ρ * R o u ^ * ⋅ ∇ * T * = − γ ( γ − 1 ) M a 2 p * ( ∇ * ⋅ u ^ * ) , (9)

where the relevant dimensionless groups are defined in

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

p * = R o γ M a 2 ρ * T * . (10)

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 = ω k ^ ,

3) steady flow,

4) subsonic flow,

5) unidirectional flow along the x ^ -axis such that u ^ = u ^ i ^ ,

6) inviscid,

7) adiabatic, and

8) negligible heat conduction.

Based on these assumptions we have neglected any influences listed by Lyman [

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

u ^ * d ρ * d x ^ * + ρ * d u ^ * d x ^ * = 0, (11)

R o u ^ * d u ^ * d x ^ * = − 1 ρ * d p * d x ^ * + x ^ * R o , (12)

R o = u c Ω c x c | Rossby number |
---|---|

M a 2 = u c 2 γ R s T c | Mach number |

γ = c p c v | Heat capacity ratio |

R o ρ * u ^ * d T * d x ^ * = − γ ( γ − 1 ) M a 2 p * d u ^ * d x ^ * . (13)

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

ρ * = C u ^ * , (14)

T * = D u ^ * γ − 1 , (15)

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

p * = R o γ M a 2 C D u ^ * γ . (16)

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

u ^ * d u ^ * d x ^ * = 1 M a 2 D u ^ * γ d u ^ * d x ^ * + x ^ * R o 2 , (17)

whose solution is

u ^ * 2 2 + 1 ( γ − 1 ) M a 2 D u ^ * γ − 1 − x ^ * 2 2 R o 2 = E . (18)

Equation (18) is an expression of Bernoulli’s theorem in a rotating framework. Inserting the boundary conditions T * ( − 1 ) = 1 , and u ^ * ( − 1 ) = 1 yields

T * = 1 u ^ * γ − 1 , (19)

1 2 ( u ^ * 2 − 1 ) + 1 ( γ − 1 ) M a 2 ( 1 u ^ * γ − 1 − 1 ) − 1 2 R o 2 ( x ^ * 2 − 1 ) = 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:

u ^ 2 − u ^ c 2 2 ︸ linear kineticenergy + c p ( T − T c ) ︸ enthalpy − ω 2 ( x ^ 2 − x c 2 ) 2 ︸ rotational kineticenergy = 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

T * = 1 − ( γ − 1 ) M a 2 2 R o 2 ( 1 − x ^ * 2 ) . (22)

Re-dimensionalizing Equation (22) and evaluating at x ^ = 0 yields the temperature distribution found by Polihronov and Straatman [

T c − T = ω 2 x c 2 2 c p . (23)

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 [^{3} evenly spaced grid points. Air was chosen as the working fluid, with a heat capacity ratio γ = 1.4 , 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

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:

1 ρ d ρ d x ^ + 1 A d A d x ^ + 1 u ^ d u ^ d x ^ = 0 , (24)

u ^ d u ^ d x ^ = − 1 ρ d p d x ^ + x ω 2 , (25)

c v d T d x ^ = − p ρ ( 1 A d A d x ^ + 1 u ^ d u ^ d x ^ ) . (26)

Invoking the ideal gas Equation (10), introducing the scaled cross-sectional area A * ( x ^ * ) = A ( x ^ ) / A c , and non-dimensionalizing 24 - 26 yields:

1 ρ * d ρ * d x ^ * + 1 A * d A * d x ^ * + 1 u ^ * d u ^ * d x ^ * = 0 , (27)

R o u ^ * d u ^ * d x ^ * = − 1 ρ * d p * d x ^ * + x ^ * R o , (28)

1 T * d T * d x ^ * = − ( γ − 1 ) ( 1 A * d A * d x ^ * + 1 u ^ * d u ^ * d x ^ * ) . (29)

Equations (27) and (29) may be solved by direct integration, and Equation (10) may be used to obtain an expression for the pressure distribution:

ρ * = C A * u ^ * , (30)

T * = D ( A * u ^ * ) γ − 1 , (31)

p * = R o γ M a 2 C D ( A * u ^ * ) γ . (32)

Solving Equation (28) requires the use of Equations (30) and (32) to obtain

( u ^ * − D M a 2 1 ( A * u ^ * ) γ − 1 u ^ * ) d u ^ * d x ^ * − D M a 2 1 ( A * u ^ * ) γ − 1 A * d A * d x ^ * − x ^ * R o 2 = 0. (33)

Equation (33) may be solved using the method of exact differentials, yielding

u ^ * 2 2 + 1 M a 2 ( γ − 1 ) D ( u ^ * A * ) γ − 1 − x ^ * 2 2 R o 2 = E . (34)

Three boundary conditions are required to evaluate constants C, D and E. If the duct area is constant (i.e. A ( x ^ ) = A c ) then A * = 1 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

A * ( x ^ * ) = ( 3 | x ^ * | + 4 ) 2 . (35)

A 1D mesh of constant grid spacing with 10^{3} grid points was generated. The solution was again computed using ANSYS CFX Ó [^{−4}. The results have been plotted in

In this section we will further generalize the above results to an arbitrarily curved passage defined by the parameterization

p ( a ) = X ( a ) i ^ + Y ( a ) j ^ + Z ( a ) k ^ . (36)

and the scaled path vector is given by p * = p / x c . The components of p may be any well-behaved functions, producing, for example, the path shown in

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

t = p ′ ( a ) ‖ p ′ ( a ) ‖ = p * ′ ( a ) ‖ p * ′ ( a ) ‖ . (37)

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:

u ^ * u ¯ * = t , (38)

where u ¯ * = ‖ u ^ * ‖ .

The following steps apply when the duct cross-sectional area is constant along the path. If we have some quantity ϕ ( p * ( a ) ) , its total derivative is

d ϕ d a = ∂ ϕ ∂ x d x d a + ∂ ϕ ∂ y d y d a + ∂ ϕ ∂ z d z d a = ∇ ϕ ⋅ p * ′ ( a ) = ( p * ′ ( a ) ⋅ ∇ ) ϕ . (39)

Furthermore, given that all quantities are defined only on the path p ( a ) , any gradient (e.g. ∇ ϕ ) is parallel to the unit tangent vector:

∇ ϕ = c d ϕ d a t , (40)

for some unknown value c. We will choose c = 1 / ‖ p * ′ ( a ) ‖ to satisfy Equation (39). Substituting Equations (38) and (40) into the governing Equations (7) and (9) yields (after some manipulation)

1 ρ * d ρ * d a + 1 u ¯ * d u ¯ * d a = 0 , (41)

1 T * d T * d a + ( γ − 1 ) u ¯ * d u ¯ * d a = 0. (42)

( u ¯ * + D γ M a 2 u ¯ * γ ) R o ‖ p * ′ ‖ d u ¯ * d a t + R o ‖ p * ′ ‖ u ¯ * 2 t ′ = 2 u ¯ * ( t × Ω * ) + 1 R o [ p * − ( p * ⋅ Ω * ) Ω * ] . (43)

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 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

( u ¯ * + D γ M a 2 u ¯ * γ ) u ¯ * a = g ′ ( a ) R o 2 , (44)

where

g ′ ( a ) = p * ⋅ p * ′ − ( p * ⋅ Ω * ) ( p * ′ ⋅ Ω * ) . (45)

Equation (44) may be solved using direct integration:

u ¯ * 2 2 + 1 ( γ − 1 ) M a 2 D u ¯ * γ − 1 − g ( a ) R o 2 = E . (46)

For a rotating passage of arbitrarily varying cross-sectional area A ( a ) , and A * ( a ) = A ( a ) / A c we must include A * ( a ) 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

u ¯ * 2 2 + 1 ( γ − 1 ) M a 2 D ( u ¯ * A * ) γ − 1 − g ( a ) R o 2 = E . (47)

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:

θ = θ ( r ) , (48)

z = ζ ( r , θ ( r ) ) . (49)

We will proceed with the parameterization r = a :

p * = a [ cos ( θ ( a ) ) i ^ + sin ( θ ( a ) ) j ^ ] + ζ ( a , θ ( a ) ) k ^ . (50)

Invoking definition 46 reveals g ′ ( a ) = a whenever θ ′ ( a ) is well-behaved over the desired range of a. Under these circumstances, Equation (47) collapses to 34, and we conclude that the flow speed u ¯ * 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:

W ˙ − Q ˙ = E ˙ st + m ˙ 2 ( h 2 + u 2 2 2 ) − m ˙ 1 ( h 1 + u 1 2 2 ) . (51)

Because the flow is adiabatic and steady we may neglect the heat transfer Q ˙ and transient energy storage E ˙ st respectively. Furthermore we recognize that m ˙ 1 = m ˙ 2 = m ˙ and insert Equation (6). With these simplifications, we have

W ˙ = m ˙ ( c p ( T 2 − T 1 ) + 1 2 ( u 2 2 − u 1 2 ) ) . (52)

In a straight, radial duct such as the one shown in

W ˙ = m ˙ ( c p T 2 + ( u ^ 2 2 + ω 2 x ^ 2 2 ) 2 2 − c p T 1 − ( u ^ 1 2 + ω 2 x ^ 1 2 ) 2 2 ) = m ˙ ( c p ( T 2 − T 1 ) + u ^ 2 2 − u ^ 1 2 2 + ω 2 ( x ^ 2 2 − x ^ 1 2 ) 2 ) = m ˙ ω 2 ( x ^ 2 2 − x ^ 1 2 ) (53)

W ˙ * = ( x ^ 2 * 2 − x ^ 1 * 2 ) R o 2 . (54)

where W ˙ * = W ˙ / m ˙ u ^ c 2 and Equation (21) has been used. Equation (53) might be rewritten in terms of a duct tip speed c = ω x ^ 2 , so that where x ^ 1 = 0 , the work transferred to/from the passage is given by

W ˙ = m ˙ c 2 . (55)

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 u ( a ) = ‖ u ¯ t + Ω × p ‖ . Substituting this expression into Equation (52) gives

W ˙ = m ˙ ( c p ( T 2 − T 1 ) + 1 2 ( ‖ u ¯ 2 t 2 + Ω × p 2 ‖ 2 − ‖ u ¯ 1 t 1 + Ω × p 1 ‖ 2 ) ) . (56)

This equation cannot be reduced any further without knowing the form of p * .

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.

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 [

M a S < 1

u 2 γ R s T 2 < 1

ω 2 x ^ 2 γ R s T 2 < 1

M a 2 R o 2 < 1

M a 2 < R o 2 . (57)

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 [

h 0 = h + u 2 2 . (58)

If the fluid is assumed to be a perfect gas, the total temperature is found by invoking Equation (6)

T 0 = T + u 2 2 c p = T + u ^ 2 + ω 2 r ^ 2 2 c p . (59)

Equation 60 is similar to Equation (17-4) in the thermodynamics text by Cengel and Boles [

In contrast, the total temperature in the rotating frame may be defined as:

T ^ 0 = T + u ^ 2 2 c p (60)

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:

u ^ 2 2 c p + T − ω 2 r ^ 2 2 c p = T ¯ 0 .

Rearranging the above equation gives

T ¯ 0 = T + u ^ 2 − ω 2 r ^ 2 2 c p . (61)

We have called T ¯ 0 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 r ^ , 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, T ¯ 0 , is. Readers familiar with turbomachinery analysis will recognize the quantity T ¯ 0 c p as the rothalpy [

The isentropic gas equations may be used to find relationships between stagnation and static pressure and density:

p ¯ 0 p = ( T ¯ 0 T ) γ / ( γ − 1 ) , (62)

ρ ¯ 0 ρ = ( T ¯ 0 T ) 1 / ( γ − 1 ) . (63)

Furthermore, we can use Equation (61) to define the ratio of stagnation to static temperature in terms of dimensionless numbers:

T ¯ 0 T = 1 + u ^ 2 − ω 2 r ^ 2 2 c p T ,

T ¯ 0 T = 1 + γ − 1 2 M a L 2 1 + γ − 1 2 ( r ^ r ^ c ) 2 M a t 2 , (64)

where M a L is a local Mach number and M a t = ω x c / γ R s T ¯ 0 is the tip Mach number. Equation 64 reduces to its stationary counterpart (Equations (16)-(17) in Ref. [

In addition, we evaluate Equation (47) at the location r ^ * = 1 to devise two useful relationships between Ma, Ro, γ , and M a t :

1 2 + 1 ( γ − 1 ) M a 2 − 1 2 R o 2 = 1 ( γ − 1 ) M a t 2 R o 2 , (65)

M a 2 R o 2 M a t 2 = T ¯ 0 T c = 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 . (66)

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

m ˙ = ρ A u ^ = p A γ R T M a L . (67)

Using property ratios 65 and 63 and simplifying yields

m ˙ = p ¯ 0 A γ R T ¯ 0 M a L [ 1 + γ − 1 2 ( r ^ r ^ c ) 2 M a t 2 1 + γ − 1 2 M a L 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (68)

We can also define the maximum possible mass flow rate for any given duct, by differentiating Equation (68) with respect to M a L and setting the result equal to 0, which yields M a L = 1 . Inserting this restriction into Equation (68) yields the critical, or choked mass flow:

m ˙ Ch = p ¯ 0 A γ R T ¯ 0 [ 2 γ + 1 ( 1 + γ − 1 2 ( r ^ r ^ c ) 2 M a t 2 ) ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (69)

We can nondimensionalize with m ˙ Ch = m ˙ Ch * ρ c A c u ^ c :

m ˙ Ch * = ρ ¯ 0 ρ c T ¯ 0 T c A * M a [ 2 γ + 1 ( 1 + γ − 1 2 r ^ * 2 M a t 2 ) ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (70)

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:

m ˙ Ch * = A * M a [ 2 γ + 1 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 ( 1 + γ − 1 2 r ^ * 2 M a t 2 ) ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (71)

Equation (71) represents the maximum possible mass flow rate at any radial location. Notice m ˙ Ch * varies with the radial co-ordinate r ^ * . If, at any location, m ˙ Ch * > 1 , 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 r ^ * :

d m ˙ Ch * d r ^ * = A * B γ + 1 2 [ 1 + γ − 1 2 r ^ * 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] − 1 r ^ * M a t 2 + d A * d r ^ * B [ 1 + γ − 1 2 r ^ * 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (72)

where we have defined the parameter B for compactness:

B = 1 M a [ 2 γ + 1 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] .

Setting Equation (72) equal to 0 yields

1 A * d A * d r ^ * = − γ + 1 2 r ^ * M a t 2 1 + γ − 1 2 r ^ * 2 M a t 2 . (73)

For the case when A * = 1 , the above equation suggests the only extrema is at r ^ * = 0 . By differentiating 73 again and inserting r ^ = 0 gives

d 2 m ˙ Ch * d r ^ * 2 = A * B γ + 1 2 M a t 2 .

Since each of the terms in the above equation are positive, the concavity of 71 is positive at r ^ * = 0 , confirming that r ^ * = 0 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 r ^ * = 0 into Equation (71) yields

m ˙ Ch,min * = 1 M a [ 2 γ + 1 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (74)

Therefore, in order to ensure the flow is not choked, we require

m ˙ * < m ˙ Ch,min * ,

1 < 1 M a [ 2 γ + 1 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] ,

1 < 1 M a [ 2 γ + 1 M a 2 R o 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] ,

2 γ + 1 M a 4 / ( γ + 1 ) > R o 2 M a t 2 . (75)

Combining Equations (75) and (65) and re-arranging results in a cumbersome inequality in terms of Ma and Ro, which has been plotted in

If the duct area varies with the equation A * = r ^ * , r ^ * > 0 (a rotating slice), Equation (73) reduces to r ^ * = i / γ M a t . 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: r ^ * = r ^ 1 * , r ^ 2 * where r ^ 1 * < r ^ 2 * . Clearly, m ˙ Ch * ( r ^ 1 * ) < m ˙ Ch * ( r ^ 2 * ) , and the maximum mass flow rate is

m ˙ Ch,min * = r ^ 1 * M a [ 2 γ + 1 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 ( 1 + γ − 1 2 r ^ 1 * 2 M a t 2 ) ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (76)

If Equation (71) is evaluated such that the flow is choked everywhere ( m ˙ Ch * = 1 ), we can formulate the critical cross-sectional area profile:

A Ch * = M a [ γ + 1 2 1 1 + γ − 1 2 M a 2 1 + γ − 1 2 M a t 2 1 + γ − 1 2 r ^ * 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] = M a t * ( M a t , r ^ * , γ ) A t * ( M a , γ ) . (77)

where we have recognized the appearance of the area ratio A / A t , which has been defined for stationary ducts:

A t * = A A t = 1 M a [ 2 γ + 1 ( 1 + γ − 1 2 M a 2 ) ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (78)

We have also introduced a modified tip Mach number,

M a t * = [ 1 + γ − 1 2 M a t 2 1 + γ − 1 2 r ^ * 2 M a t 2 ] ( γ + 1 ) / [ 2 ( γ − 1 ) ] . (79)

Equation (78) has been tabulated for many values of Ma and γ in many engineering texts such as Ref. [

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

et al. [

The authors gratefully acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC).

The authors declare no conflicts of interest regarding the publication of this paper.

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

Roman Symbols

A Duct or passage cross-sectional area

a Independent parameter

A Ch * = A r * / A t * Critical cross-sectional area profile

A t * = A / A t Area to throat area ratio in a stationary passage

C, D, E Constants of integration

c_{p} Isobaric heat capacity

c_{v} Volumetric heat capacity

E ˙ s t Transient energy storage in a control volume

h = ε + p / ρ Specific enthalpy

i = − 1

i ^ , j ^ , k ^ Unit vectors aligned with the x; y; and z axes, respectively

m ˙ Mass ow rate

m ˙ C h Maximum (choked) mass ow rate

m ˙ C h , min 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

R_{8} Specific ideal gas constant

T 0 = T + u 2 2 c p Total temperature

t Unit tangent vector to a parametric curve P*

u Velocity

u ¯ Flow speed along along a constrained path

x Position vector

X, Y, Z Components of position vector

x, y, z Cartesian co-ordinates

Greek Symbols

γ = c p c v Ratio of speci_c heats

∇ Gradient operator

ε speci_c internal energy

ρ fluid density

ϕ Arbitrary scalar or vector quantity

ϕ ¯ 0 Stagnation quantity

Ω Angular velocity of rotating frame

Dimensionless Groups

M a 2 = u c 2 γ R s T c Global Mach number

M a S = u 2 γ R s T 2 Shroud Mach number

M a L = u ^ γ R s T Local Mach number

M a t = Ω c x c γ R s T ¯ 0 Tip Mach number

M a t * Modified tip Mach number ratio

R o = u c Ω c x c Rossby number

Superscripts

* Non-dimensional quantity

Subscripts

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

Other Symbols

' Derivative of single-variable function

^ Quantity in non-stationary frame

Acronyms

1D One-dimensional

CFD Computational Fluid Dynamics

RHVT Ranque-Hilsch Vortex Tube

RHS Right Hand Side