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

∇ ⋅ ( 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 [18] .

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 Table 1.

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

p * = R o γ M a 2 ρ * T * . (10)

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   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 [1] :

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 [19] 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 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 Figure 2. A maximum error of 0.03% was observed between Equation (18) and the CFD velocity profile, and a maximum error of 5 × 10 − 4 % was observed between Equation (15) and the CFD temperature profile.

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. [12] .

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

A 1D mesh of constant grid spacing with 10³ grid points was generated. The solution was again computed using ANSYS CFX Ó [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 (34) and the CFD results while a maximum error of 0.004% was observed between Equation (31) and the CFD results.

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.

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, M a S , should be less than 1:

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 [21] :

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 [3] , 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:

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 [7] . Notice that the total temperature and stagnation temperature are not equal in general; i.e. T ^ 0 ≠ T ¯ 0 .

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. [21] ) when ω = 0 .

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)