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Unsteady flow modelling of a pressure wave supercharger A Fatsis, N G Orfanoudakis, D G Pavlou, A Panoutsopoulou and N Vlachakis Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 2006 220: 209 DOI: 10.1243/095440706X72628 The online version of this article can be found at: http://pid.sagepub.com/content/220/2/209
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Unsteady flow modelling of a pressure wave supercharger A Fatsis1*, N G Orfanoudakis1, D G Pavlou1, A Panoutsopoulou2, and N Vlachakis1 1Technological University of Chalkis, Department of Mechanical Engineering, Psachna Evias, Greece 2Hellenic National Defence Systems S.A., Greece The manuscript was received on 6 January 2005 and was accepted after revision for publication on 21 September 2005. DOI: 10.1243/095440706X72628
Abstract: Pressure wave machinery constitutes a promising device in a wide variety of applications such as superchargers in automobile engines and topping devices in gas turbines. A very attractive feature is the built-in exhaust gas recirculation which provides a substantial reduction in NO emissions of the engine. The present work presents a simple numerical model x based on shock tube flow, accounting for viscous and thermal losses inside the rotor of the supercharger as well as leakage losses at the extremities of the rotor. It is validated against available experimental data of shock tube flow and on a three-port wave rotor, called a pressure divider. The model is then applied on a configuration suited for automobile engine supercharging. Comparisons with experimental data show that the present method can analyse the basic flow patterns inside the rotor, providing a useful tool for pressure wave supercharger flow predictions. Keywords: pressure wave supercharger, wave rotor, internal combustion engine, unsteady flow, shock tube flow
1 INTRODUCTION Pressure wave superchargers or wave rotors or ComprexA are rotating devices in which energy is transferred between two gaseous fluids by short-time direct contact of the fluids in slender flow channels. They are composed of two concentric cylinders between which radial straight planes are arranged, giving rise to long channels of constant cross-section [1]. Lateral non-rotating perforated flanges (stators) are mounted upstream and downstream of the rotor. Through the openings of the stators, which are commonly called inlet and outlet ports, the air and the hot gases enter and exit the rotor. Different applications of wave rotors can be obtained depending on the following parameters:
(c) the aerothermodynamic quantities specified at the inlet and outlet of the ports; (d) the direction of the inflow and outflow at each port.
Greece.
[email protected]
Initially, wave rotors were designed to supercharge internal combustion engines. In such applications, the ports connect the rotor to the fresh air intake, to the exhaust pipe, and to the inlet and outlet of the combustion chamber of the engine, as illustrated in Fig. 1. Automobile applications of the pressure wave supercharger over a wide range of car and truck diesel engines showed a fast response to changes in the engine load, resulting in almost instantaneous availability of maximum torque [1]. The ability of the wave rotor to produce high pressure ratios even at low engine speed permits the engine designer to provide high torque back-up and thus induce the driver frequently to drive at low speed. This results in fuel economy and low noise levels. The built-in exhaust gas recirculation feature due to the fact that exhaust gas and fresh air are in direct contact helps to reduce the emissions of oxides of nitrogen (NO ), x as reported in references [2] and [3].
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(a) the number of ports upstream and downstream of the rotor; (b) the dimensions of the ports; * Corresponding author: Technological University of Chalkis, Mechanical Engineering Department, 34400 Psachna Evias,
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The present contribution aims to present a general and accurate numerical tool suited for the analysis of unsteady flows encountered inside different types of wave rotor. After a brief description of the numerical method, the model is validated against experimental data for shock tube flow and for the flow inside a three-port pressure divider. Then the model is implemented to predict the flow inside a reverse-flow wave rotor that can be used as a supercharger of internal combustion engines. Comparisons with measurements show that the present method gives accurate results for the cases examined and that it is able to capture the basic unsteady flow phenomena inside the rotor.
2 NUMERICAL METHOD Fig. 1 Four-port wave rotor used to supercharge an internal combustion engine: 1, air at atmospheric conditions; 2, compressed air to the combustion chamber; 3, hot gases from the combustion chamber; 4, hot gases and air to the exhaust
Another application is the pressure divider. This type of wave rotor is generally equipped with three ports. It is used to split the inlet flow into two flows – one at a higher pressure and the other at a lower one [4, 5]. Wave rotors have also been proposed as topping devices for gas turbines [6, 7]. Main examples are the four-port [6] and the five-port wave rotor [7]. Recent studies propose wave rotor configurations allowing compression, combustion, and expansion processes of the fluid within one cycle of operation [8]. The flow inside the rotor is unsteady. It is dominated by propagation of compression and expansion waves which interact with each other and reflect on the solid walls of the upstream and downstream stators as well as on the inflow and the outflow boundaries. For all the above types of wave rotor, the method of characteristics was used to calculate the flow inside the rotor channels. Such an approach does not account for losses incurred during the actual operation of the rotor. One-dimensional methods based on the Euler equations were also used, including modelling for losses that occur inside the rotor [9, 10]. Two-dimensional methods are rather time consuming and are mainly suited to detailed study of specific regions of interest of the flowfield inside the rotor, such as finite opening/ closing effects and shock wave–boundary layer interaction [11].
2.1 Assumptions In the existing pressure wave superchargers manufactured so far it is observed that the length of each wave rotor channel is much larger than its height and width [1–3]. This justifies the approximation of the flow inside a channel formed by two consecutive blades with the flow inside a shock tube. This approach simplifies the real three-dimensional and unsteady flow inside the rotor channels with a onedimensional unsteady flow inside a shock tube. With this simplification, centrifugal forces and secondary flows occurring inside the rotor channels are ignored. A two-dimensional analysis method, though, could clarify the limits of applicability of one-dimensional methods. As a first approximation, this approach can be considered realistic. Nevertheless, pressure losses, viscous phenomena, and leakage losses exist in the flow inside a real wave rotor. The present model takes into account these losses by using semi-empirical correlations found in the literature [9, 10]. Moreover, it is assumed that, when one channel arrives at the edge of one of the stator plates, the flow from or towards the channel accelerates instantaneously. It is the same as assuming that in a shock tube the membrane separating two gases of different pressures is located at one of the tube extremities. After the membrane rupture, a shock wave, a contact interface, and an expansion fan are propagating inside the tube [12, 13]. 2.2 Governing equations Following the aforementioned assumptions, the basic model consists of the Euler equations describing
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the unsteady flow inside a one-dimensional channel of constant cross-section. These equations in conservative form, expressing the conservation of mass, momentum, and energy [14], are written as qU qF(U ) + =S qt qx
integration method Dt=
CFL Dx 2p ∏ NITER×RPM/60 |u|+c
(2)
2.5 Boundary conditions (1)
The conservative variables and fluxes, U and F respectively, can be found in reference [12]; S is the source term vector and will be analysed in a later section.
2.3 Spatial discretization Experience using different numerical schemes showed that upwind schemes are well suited for compressible flows including discontinuities such as shock waves [14]. The second-order accurate scheme of Roe was chosen for the space discretization of the system of partial differential equations (1) [15]. This scheme very accurately captures the discontinuities such as shock waves that are expected to be present in such a flowfield. The computation of the numerical flux of the Roe scheme, as well as the different flux limiter functions, was performed according to reference [16].
2.4 Time integration Since the behaviour of the unsteady flow inside wave rotor channels is of interest in the present study, the time integration scheme should offer high accuracy. For this reason, the four-step Runge–Kutta scheme is an attractive choice [17]. Studies show that the above scheme is second-order accurate for nonlinear equations. Its stability limit extends up to CFL∏2√2, where CFL stands for the Courant– Friedrichs–Lewy number characterising the extent of stability of numerical schemes, according to the numerical stability theory that can be found in reference [14]. It is important to underline that, in the case of unsteady flows, a constant time step Dt should be used for all points of the numerical domain. This time step should link the numerical computation to the physics of the flow phenomenon. The numerical time step between two consecutive circumferential rotor positions is linked to the wave rotor angular speed and the number of iterations per rotation (NITER), specified by the user. This time step has to respect the CFL stability condition of the numerical D00305 © IMechE 2006
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A general mathematical model for the analysis of flows inside wave rotors should be able to handle any possible combination of boundary conditions at the extremities of the rotor channels. During the operation of a wave rotor it is possible to have different boundary conditions: subsonic or supersonic flow conditions towards the inlet and outlet of the computational domain (rotor channel). According to the theory of characteristics given in references [12] and [14] the numerical model employs different boundary conditions depending on the nature of the flow in the extremities of the wave rotor channels. In particular, when the inflow is subsonic to the right there are two incoming characteristics corresponding to the entropy and the downstream propagating pressure wave. Consequently, two physical quantities have to be specified at this boundary (total pressure and total temperature). When the outflow is subsonic to the right, one characteristic enters the domain corresponding to the upstream-propagating pressure wave. In this case the outlet static pressure is the chosen specified physical quantity. Supersonic inflow corresponds to three incoming characteristics, therefore three physical quantities have to be specified at the inlet boundary (total pressure, total temperature, and the Mach number). In the case of supersonic outflow there are no incoming characteristics and therefore there is no physical quantity specified at this boundary. All boundary conditions implemented use the compatibility equations method described in reference [14]. The boundary conditions are of reflecting type, simulating infinite reservoirs with constant flow conditions connected to the rotor inflow and outflow ports. Under these boundary conditions, when moving shock waves reach the rotor extremities, they reflect back to the rotor channel and interact with other.
3 EVALUATION OF SOURCE TERMS The basic numerical model as presented above was improved to take into account the most important loss mechanisms during the operation of an actual wave rotor such as viscous losses and losses due to axial and radial leakage [5]. The right-hand side of Proc. IMechE Vol. 220 Part D: J. Automobile Engineering
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system of equations (1) can be written as S=S
visc
+S
3.2 Leakage losses (3)
leak
3.1 Viscous losses In general, the flow pattern in the wave rotor channels is turbulent, according to flow visualizations performed in reference [18]. Viscous losses affect the momentum and the energy equation of system (1). They are of the form S
visc where
=(0 s s )T 2 3
t s =−4 2 D h and
Leakage occurs only at the extremities of each of the rotor channels, in the axial gap between the rotor and the stators. The present model assumes that leakage losses are included only at the first and last computational cells of the domain. The leakage source terms affect the mass and energy equations [5] S
=(s1 0 s1 )T leak 1 3 where s1 and s1 are given in reference [5]. 1 3 4 VALIDATION
k T −T w s =− t 3 m u according to reference [13]. The shear stress, t, and wall temperature, T , are w calculated according to the correlation proposed in references [18] and [13] respectively.
A first validation series concerns the implementation of the model in shock tube flow and the comparison of the results obtained with experimental ones found in the literature [5]. According to this test case, a shock tube, which has a ratio of length to hydraulic diameter of 400, is closed at both ends. At x/L=x 0 there is a membrane separating two gases; the one on the left is at higher pressure and the pressure ratio is 9.8. Figures 2(a) and (b) show the comparison
Fig. 2 Non-dimensional (a) velocity and (b) mass flow distribution 15 ms after the membrane rupture Proc. IMechE Vol. 220 Part D: J. Automobile Engineering
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between experimental data obtained in reference [5] and numerical predictions of the present model at time t=15 ms after membrane rupture. Dashed lines illustrate results from an inviscid flow computation. Continuous lines indicate predictions incorporating viscous and thermal losses. There is very good agreement between experimental data and predictions using turbulent shear stress modelling [Fig. 2(a)]. The agreement is still very good in non-dimensional mass flow predictions [Fig. 2(b)], except for the region of contact discontinuity at 0.65∏x/L∏0.75. In that region, inviscid flow calculations show an abrupt drop, as expected [Fig. 2(b)]. The improved model seems to spread the discontinuity, coming closer to experimental points, but it cannot fully match the measurements. Another comparison between experimental data and numerical predictions for the same test case [5] can be seen in Fig. 3, where the static pressure at a given point inside the tube is given as a function of time. When the shock passes from this point, an abrupt jump in pressure is observed. After a time interval t0.10, inviscid flow computations (illustrated with dashed lines) predict a constant value of static pressure, while experiment and predictions based on turbulent flow correlations indicate an increase in static pressure. The comparison between experimental data from reference [5] and predictions using the correlations of unsteady shock tube flows is still very good, a fact that reinforces the validity of the developed numerical model. A second validation test refers to the calculation of the flow inside the three-port through-flow wave rotor or ‘pressure divider’. This wave rotor was manufactured and instrumented to validate numerical models aiming to analyse its flow pattern [4]. In order to accommodate measuring devices, external ducts
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were connected to the inlet and outlet ports of the rotor. Standard orifice meters were placed in the ducts in order to measure the mass flows passing through the inlet and outlet ports. Pitot tubes and thermocouple probes were also placed in the inflow duct to measure stagnation pressure and temperature respectively. The static pressure at the outlet ports was measured by placing wall static pressure taps around the circumference of the ducts. The experimental data of the unsteady pressure distribution at axial rotor positions, x/L=0.025, 0.50, 0.975 were also reported in reference [4]. Pressure transducers were placed in a rotor channel at the above axial locations, measuring the variation in pressure with time. The signals from these transducers were sent to stationary external amplifiers and data recording equipment. The numerical simulations assume that initially (t=0) the rotor is filled with a fluid of uniform pressure and density and zero velocity. This assumption is used only as an initial condition and does not affect the numerical results. The convergence history of the time-accurate calculations is shown in Fig. 4. The upper graph of this figure shows the static pressure distribution at the axial location x/L=0.025 in terms of the circumferential coordinate during five rotations. In the lower part, the distribution of the first rotation is compared with the distribution of the fifth rotation. The slight differences in the beginning (0∏h∏p/5) can be attributed to the initialization procedure. It is also observed that, during the last two or three rotations, the static pressure profile repeats itself, which means that the calculations are well converged. The period of the phenomenon is 2p rad. The factor that imposed the periodicity of the flowfield is the geometry of the rotor.
Fig. 3 Static pressure distribution at x/L=0.66 as a function of non-dimensional time D00305 © IMechE 2006
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Fig. 4 Convergence history of the unsteady flow computations
Figure 5 presents a comparison between the predictions obtained using the present numerical model and experimental data from reference [4] illustrated by circles. The dotted lines present inviscid flow calculations, the long dashed lines present the results obtained, including viscous and thermal losses, and the continuous lines present the predictions of the final model, which also include leakage losses at the extremities of the rotor (improved model). Inviscid flow predictions show a larger amplitude of oscillation for the three axial locations. This behaviour is expected because upwind schemes (such as the one used in this model) include just the minimum numerical damping provided by the numerical flux limiters [16]. Predictions including viscous and thermal losses give better results than the inviscid flow predictions when compared with the experimental data. The amplitude of pressure oscillations of the model including viscous and thermal losses is reduced owing to the dissipation resulting from the addition of the loss modelling. Ultimately, results obtained with added leakage source terms improve the numerical predictions. Although leakage source terms are included only at the first and last computational cells, they affect the
flowfield even at x/L=0.5. It can be seen that the agreement between the final (improved) model and experimental data is very satisfactory for the three axial locations examined. There are some regions, though, like the one corresponding to the opening of the inflow port on the left (x/L=0.025, hµ[2, 3] rad), where the agreement is not so good. These differences occur owing to the fact that the present model supposes that the flow is accelerated instantaneously when a rotor channel arrives in front of an inflow port. In reality there are losses associated with the inflow and outflow phenomena (finite opening time effects) that are not taken into account in the present numerical model. Approximate loss models already existing in the literature may improve the results [5]. Figure 6, obtained by using the complete model, presents a plot of the static pressure nondimensionalized by the total pressure at the inlet port (reference conditions) during one rotation. This figure shows an unrolled projection of the three-port wave rotor. The abscissa corresponds to the nondimensional x/L distance and extends from 0 to 1, while the ordinate corresponds to the circumferential coordinate and extends from 0 to 2p. It is supposed that the wave rotor rotates towards the upper part of
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Fig. 5 Comparison between numerical predictions and experimental data for the three-port wave rotor
Fig. 6 Static pressure field non-dimensionalised by the total inlet total pressure for the three-port wave rotor D00305 © IMechE 2006
Fig. 6. The operation begins at the lower part when the port of low pressure P opens instantaneously. b An expansion fan is created that propagates to the left. Arriving at the solid wall upstream of the rotor, the fan hits it and reflects back to the right. At that moment the inflow port at pressure P opens and a m shock wave is created when the high-pressure inlet gas comes into contact with the low-pressure gas in the expansion fan. This shock propagates to the right, bouncing between the left and right solid walls. At this point the port at higher pressure P opens and h a percentage of the flow exits the rotor. When this port closes, an expansion fan is created and reacts with the shock, which has almost arrived at the right end of the tube. The shock reflections on the left and right solid walls of the rotor continue and the amplitude of the static pressure oscillations attenuates up to the end of the rotation where the flow becomes almost axially uniform. Then a new rotation is about Proc. IMechE Vol. 220 Part D: J. Automobile Engineering
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to be launched. The above flowfield analysis is consistent with the observations of other researchers as reported in references [4] and [5].
5 REVERSE-FLOW FOUR-PORT WAVE ROTOR This type of wave rotor is currently used in automobile applications [1–3], and experimental data on the subject have been published in reference [19] using the same experimental techniques as for the case of the three-port wave rotor described in the previous section. Figures 7(a) and (b) show predicted total pressure and total temperature inside the rotor, non-dimensionalized by the total pressure and temperature at the inlet of port 1, during one complete rotation. The rotor is supposed to rotate from the lower to the upper part of Figs 7(a) and (b). The abscissa is the non-dimensional distance x/L extending from 0 to 1, and the ordinate corresponds to the circumferential coordinate h extending from 0 to 2p. The process in this cycle begins at the lower part of Figs 7(a) and (b) when port 4 towards the exhaust pipe opens and hot gases with a percentage of relatively cold, compressed air exit the rotor. As a result, an expansion wave is generated that propagates to the left, hitting the upstream stator walls and then reflecting to the right as a compression wave. At this moment, port 1 opens and, owing to the large pressure difference, fresh air enters, scavenging the rotor cells and pushing the
remaining gases towards the exhaust pipe port 4. As port 4 closes, the compression wave that was initiated when port 1 was open hits the downstream stator wall and reflects to the left as a shock wave. This shock wave front progresses faster than the speed of sound before it, and it is simultaneously compressed and accelerates the air towards the left. At that moment, port 2 opens and hot gases from the combustion chamber of the engine enter the rotor, creating a strong shock wave propagating to the left, compressing and pushing air and remaining gas towards the combustion chamber (port 3) [Fig. 7(a)]. When the front between the compressed air and hot gases reaches port 3, this port closes, preventing exhaust gas recirculation into the engine. Some of the air that has been contaminated remains in the rotor cells. As port 3 towards the combustion chamber closes, a compression wave is created, pushing the remaining gas and compressed air towards exhaust pipe port 4. This remaining air gives the energy to scavenge the cells, that is, to replace the exhaust gas with air. From Fig. 7(b) it can be seen that hot gases mostly remain in the right part of the rotor, corresponding to ports 2 and 4, while the air remains in the left part of the rotor, corresponding to ports 1 and 3. Other researchers have made similar observations [1–3]. The wave processes described correspond to a ‘perfectly tuned’ pressure wave supercharger according to the engine demands. In reality, a supercharger has to function over a wide engine operating range
Fig. 7 (a) Total pressure non-dimensionalized by the pressure at port 1 and (b) the total temperature non-dimensionalized by the total temperature in the port 1 field for the reverseflow four-port wave rotor: 1, air at atmospheric conditions; 2, compressed air from the combustion chamber; 3, hot gases to the combustion chamber; 4, hot gases and air to the exhaust Proc. IMechE Vol. 220 Part D: J. Automobile Engineering
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from very low up to very high rotating speeds. The pressure wave supercharger successfully achieved this wide-range speed operation by using additional stator ports, called ‘pockets’, which are not connected to any duct [1, 3]. The present model does not account for the effect of ‘pockets’ at off-design conditions. Detailed comparisons between predicted and measured static pressure distribution at axial locations x/L=0 and x/L=1 are presented in Fig. 8. At x/L=0, predictions match experimental data obtained from reference [19]. The measurement technique applied was similar to the one described in section 4. Small discrepancies occur at circumferential locations where hµ[p, 3p/2] rad, corresponding to port 1, and where hµ[7p/4, 2p] rad, corresponding to the closing of port 3. At x/L=1, differences between experimental data and predictions occur at circumferential locations where hµ[0, p/4] rad, corresponding to the opening of port 4, and where hµ[7p/4, 2p] rad, corresponding to the solid wall area. The comparison can be considered good taking into account the complexity of the unsteady flow inside the rotor channels and the fact that the
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experiment was carried out on a real engine. Yet, incorporating a loss model for the finite closing of ports effect could improve the accuracy of numerical predictions in the circumferential positions corresponding to the closing of inflow/outflow ports.
6 CONCLUSIONS A simple one-dimensional model, able to analyse unsteady flows inside multiple-port wave rotors, has been presented. Viscous and thermal loss modelling was applied by using correlations of unsteady turbulent flows inside shock tubes instead of the correlations based on steady or laminar flows that have been used by many researchers in the past. A leakage loss model was also added in the first and last computational cells to simulate mass flow losses at the extremities of the rotor channels. This model was validated for shock tube flow and for a threeport wave rotor against experimental data available in the literature. Then, the model was applied on a reverse-flow four-port pressure wave supercharger suited for internal combustion engines. The numerical results obtained showed that the model is able to
Fig. 8 Comparison between computed and measured static pressure distributions for the reverse-flow wave rotor D00305 © IMechE 2006
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predict the unsteady flowfield features inside the rotor channels, being in agreement with measurements and observations of other researchers. Finite opening–closing effects may improve predictions, simulating more accurately the phenomena occurring when a rotor channel approaches an open port or leaves a port and approaches the solid wall of a stator. Two-dimensional simulations in reference [11] can give an insight into the issue and may clarify the applicability limits of one-dimensional prediction methods. Including in the model the effect of pockets in the stator walls can lead to more realistic prediction of the performance map of pressure wave superchargers when integrated in internal combustion engines. That would make it possible to study the behaviour of the wave processes inside the rotor at off-design conditions.
REFERENCES 1 Berchtold, M. Supercharging with Comprex. VKI Lecture Series 1982-01 on Turbochargers and Related Problems, 1982. 2 Mayer, A., Pauli, E., and Gyrax, J. ComprexA supercharging and emissions reduction in vehicular diesel engines. SAE paper 900881, 1990. 3 Janssens, J. P. Simulation of the unsteady nonhomoentropic flow in pressure wave machinery using the mass flow defect concept, PhD Thesis, K.U. Leuven, 1992. 4 Wilson, J. and Fronek, D. Initial results from the NASA–Lewis wave rotor experiment. AIAA paper 93-2521, 1993. 5 Kentfield, J. A. C. Nonsteady, one-dimensional, internal, compressible flows, 1993 (Oxford University Press). 6 Fatsis, A. and Ribaud, Y. Thermodynamic analysis of gas turbines topped with wave rotors. Aerospace Sci. and Technol., 1999, (5), 293–299. 7 Resler, E., Mocsari, J., and Nalim, R. Analytic design methods for wave rotor cycles. J. Propulsion and Power, September–October 1994, 10(5). 8 Nalim, R. Preliminary assessment of combustion modes for internal combustion wave rotors. AIAA paper 95-2801, 1995. 9 Paxson, D. E. Comparison between numerically modelled and experimentally measured loss mechanisms in wave rotors. J. Propulsion and Power, 1995, 11(5), 908–914. 10 Paxson, D. E. An improved numerical model for wave rotor design and analysis. NASA technical memorandum 105915, 1993. 11 Welch, G. E. Two-dimensional computational model for wave rotor flow dynamics. ASME paper 96-GT-550, 1996.
12 Shapiro, A. H. The dynamics and thermodynamics of compressible fluid flow, 1958 (John Wiley). 13 Weber, H. Shock wave engine design, 1995 (John Wiley). 14 Hirsch, C. Numerical computation of internal and external flows, vols I and II, 1991 (John Wiley). 15 Roe, P. L. Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mechanics, 1986, 18, 337–365. 16 Yee, H. C. A class of high-resolution and implicit shock-capturing methods. VKI Lecture Series 1989-04 on Computational Fluid Dynamics, 1989. 17 Fatsis, A. Numerical study of the 3D unsteady flow and forces in centrifugal impellers with outlet pressure distortion, PhD Thesis, University of Gent – von Karman Institute, 1995. 18 Mentre´, P., Juillen, J., and Brevaux, P. Calcul approche´ des couches limites et des transferts de chaleur aux parois d’une tube a` choc. ONERA TP 647, La Recherche Ae´rospatiale 126, September– October 1968. 19 Shreeve, R. P. and Mathur, A. (Eds) Proceedings of ONR/NAVAIR Wave Rotor Research and Technology Workshop, Naval Postgraduate School, Monterey, California, 1985.
APPENDIX Notation c D h F k L P RPM S t T u U x
speed of sound hydraulic diameter flux vector thermal conductivity length of the blade channel static pressure rotor angular speed (r/min) source term vector time static temperature flow velocity conservative variables vector distance along the axis of the rotor
Dt h m r t
time step circumferential coordinate (rad) dynamic viscosity density shear stress
Subscripts leak visc w
leakage losses viscous losses wall conditions
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