16. CHARACTERISTICS OF SHOCK-WAVE UNDER LORENTZ FORCE AND ENERGY EXCHANGE

Transcription

1 16. CHARACTERISTICS OF SHOCK-WAVE UNDER LORENTZ FORCE AND ENERGY EXCHANGE H. Yamasaki, M. Abe and Y. Okuno Graduate School at Nagatsuta, Tokyo Institute of Technology 459, Nagatsuta, Midori-ku, Yokohama, Jaan Abstract. Results of analytical study on characteristics of the suersonic flow under various conditions are resented. The imosed conditions are the energy addition to the flow, the energy extraction and the Lorentz force acting as acceleration or deceleration of the flow. Relations for shock strength, Mach number and total ressure loss were derived from the one-dimensional simle analytical model. The relations are similar to those of Rankine-Hugoniot, but they include the energy exchange and the MHD effects. The relations can exlain observed henomenon of the shock weakening by heat addition. It can also exlain the measured relation between the enthaly extraction and the isentroic efficiency of the disk MHD generator and the aearance of shock wave. 1. Introduction Recently, the control of shock wave by heat addition into the suersonic flow or by an alication of a magnetic field has received interest in the field of suersonic vehicle [1-]. The concet of shock wave control by heat addition is based on exerimental observation of anomalous henomena such as the shock weakening, the change of Mach angle and the shock acceleration. In order to exlain these henomena, numerical and exerimental studies were carried out [1-4]. Results of them suggest that the observed anomalous henomena could be ascribed to thermal effect (=gas heating effect) [3-4]. The control of suersonic flow by magnetic field is an extension of MHD ower generation technology. A numerical study [1] was made to know ossibility of the control of suersonic flow at an intake of scramjet engine. And, it is ointed out that the osition of shock intersection oint shifts ustream with an increase of magnetic field and that total ressure loss increases with a magnetic field. The behavior of suersonic flow under a magnetic field is also imortant in the MHD ower generation. In the ast, a high enthaly extraction was accomanied by an aearance of shock wave due to the Lorentz force. However, a recent exerimental study at Tokyo Institute of Technology has shown for the first time that the high enthaly extraction is ossible without an aearance of shock wave. As described here, the knowledge of suersonic flow under the heat addition, the heat removal (enthaly extraction) and the magnetic field (Lorentz force), is very imortant from the viewoint of both the flow control and the efficiency of MHD ower generation. In this aer, an analytical study was carried out in order to exlain the shock behavior under various conditions without doing a comlex numerical simulation. The imosed conditions are the energy addition to the flow, the energy extraction from the flow and the Lorentz force acting as an accelerating force or a decelerating force. Relations for shock strength, Mach number and total ressure loss were derived from the onedimensional simle model. They were adated to the suersonic flow inside a channel with constant area along the flow direction.. Derivation of Shock-Relation for the Flow with Energy Exchange and Lorentz Force Rankine-Hugoniot shock relations are well known as equations that relate fluid-dynamical roerties behind the shock wave with those before the shock wave. The relations can be alied to the flow without energy exchange and the Lorentz force. But, in the resent aer, an interest is focused on shock relations for the channel flow under energy exchange and the Lorentz force. Shock relations under these conditions were derived. In order to derive the relations, the ordinal one-dimensional basic equations were used. They are one-dimensional continuity equation, momentum equation and energy equation, and they can be written as follows: d ( ρua) = 0 Mass conservation d A {( ρu + ) A} = + A( j B) x (1) () 105

2 Energy Equation ρu d u c = j E T + A A (3) ositive for energy addition from the outside into the flow and negative for energy extraction from the flow. The negative EA is known as the enthaly extraction ratio (=E.E) in the MHD ower generation. Here, ρ, u and A denote a gas density, a flow seed and an area of channel, resectively. P and T are a gas static ressure and a static temerature, resectively. J is a current vector and B is a magnetic flux density. E is an electric filed vector and c is a gas secific heat at constant ressure. In the momentum equation (), the Lorentz force j B is included, and the energy exchange between the flow and the outside is exressed by J E in the energy equation (3). These equations were alied to the control volume shown in Fig.1. Here, we secify fluid-dynamical roerties of the flow at the inlet and those at the exit by subscrit 1 and, resectively. Fig.1. Flow calculation model The basic equations were integrated from the surface 1 to the surface. In the integration, A was assumed constant along the flow direction. Furthermore, the Lorentz force j B and the energy addition J E were relaced by their satially averaged non-dimensional arameters S and EA, resectively, and they are defined as follows: < j B > L S = < ρu > < j E > AL EA = < ρuac T 0 > (4) (5) Here, < > denotes an average along the distance L. The non-dimensional arameter S is known as the interaction arameter and the sign of S is negative when the Lorentz force acts ustream and it is ositive for the Lorentz force directing downstream. EA defined by Eq. (5) is the energy addition. It is The derived shock relations Integrating the basic equations from the surface 1 to the surface, we can derive relations between fluid roerties at the surface 1 and those at the surface. Tyical two relations are written below. 1 1 M 1 ( κ 1)M1 + = (1 + EA) 1 M ( κ 1)M + ( κ+ 1) 1 ( 1) 0 M 1 ( κ 1)M + κ = (1 + EA) 01 M ( κ 1)M1 + (6) (7) Here, P 0 is a total gas ressure and is a ratio of secific heat. Eq. (6) exresses the shock strength for M <1, and total ressure loss along the channel can be estimated from Eq. (7). Mach number M included in the Eqs. (6) and (7), is given as a function of M 1, EA and S. When the energy addition EA is zero, these equations agree with the Rankine- Hugoniot relations. The authors would like to call these relations MHD Rankin-Hugoniot relations or Rankin-Hugoniot-Yamasaki relations and they are alicable to the flow with and without energy addition and Lorentz force. 3. Results of Calculation 3.1 Flow Control by Heat Exchange At first, behavior of the channel flow with heat exchange and without the Lorentz force is described. In this case, the heat addition term EA in the above equations becomes ositive for heating the flow from the outside and negative for cooling, resectively. Figure -(a) shows calculated static ressure ratio P /P 1 for three inlet Mach numbers M 1 of 1.5,.5 and 3.5. We notice in this figure that there are two solutions when energy addition EA is given, and for examle, two solutions Sol.1 and Sol. exist for EA=0 and M 1 =1.5. Sol.1 and Sol. are the suersonic solution and the subsonic one, resectively, and they corresond to two solutions calculated from the Rankine-Hugoniot relations. The suersonic solution Sol.1 is a trivial one and it indicates that the static ressure P is ket constant 106

3 (=P 1 ) for the flow without heat addition. The subsonic solution Sol. reresents the shock wave solution and the static ressure ratio P /P 1 denotes the strength of shock wave. at the heat addition of 0.65 is 1.0 for M 1 =.5 and the flow is choked. Therefore, if gas heating exceeds 0.65, the flow with M 1 less than.5 does not exist at the channel inlet. Fig.-(a). Effect of heat addition on static ressure ratio for S=0 Fig.-(a) indicates that with an increase of heat addition EA, the static ressure ratio of subsonic solutions decreases. For examle, the static ressure ratio P /P 1 for M 1 =.5 decreases with gas heating and it becomes about 4. at EA=0.65. On the other hand, the ratio P /P 1 for suersonic solution increases with gas heating and it becomes also 4. for the same EA. Thus, it is noted that in the case of M 1 =.5, any solutions can t be found for larger EA than This can be exlained as follows. One-dimensional equation which exresses a change of Mach number along the flow, is exressed by κ M 1 d M ( 1+ κ M ) 1 dt = 0 (8) M 1 M T0 Here, T 0 is a total gas temerature. In deriving this equation, the area of the channel A was assumed constant and the Lorentz force was neglected. Eq. (8) indicates that dt 0 / has a similar role to that of da/. In the convergent and divergent nozzle, flow is accelerated from subsonic to suersonic. In this case, a throat corresonds to a singular oint mathematically. Thus, da/ should be zero at the throat where flow Mach number becomes 1. Considering this henomenon, we can know that when the flow Mach number becomes equal to 1, dt 0 / should become zero. In other word, heating or cooling of gas should be terminated at a location, where the flow Mach number becomes 1.0. In fact, it can be seen from Fig.-(b) that Mach number M Fig.-(b). Effect of heat addition on Mach number M for S=0 It is noted in Fig.-(a) that with an increase of gas heating, the strength of shock wave (=P /P 1 ) is weakened and Mach number behind shock wave M is increased (Fig.-(b)). The weakening of shock wave has been ointed out in reference (3) and (4), and it is suggested that the weakening occurs when the gas heating exists in the flow. As shown here, the resent simle model can also exlain the weakening of shock wave by the heat addition. Furthermore, it is suggested that the acceleration of the flow behind shock wave may be related with the shock acceleration found in reference (3). 3. Flow Control by Negative Lorentz Force In this section, the flow control by the Lorentz force is described. Static ressure ratio P /P 1, Mach number M and total ressure ratio P 0 /P 01 were calculated as a function of interaction arameter S. Here, the interaction arameter was selected negative. The negative interaction arameter indicates that the Lorentz force j B acts toward negative x-direction. In order to aly the negative Lorentz force, there two methods; 1) alying current from an external electrical ower suly, ) use of current induced by an electromotive force inside the flow. In alying current from the external ower suly, the current should be alied in the negative y-direction (Fig.1) to imose the negative Lorentz force on the flow. The current induced by the electro-motive force flows automatically along the negative y-direction, and as a result, the Lorentz force j B acts toward the 107

4 negative x-direction. This corresonds to MHD ower generation. The alication of the Lorentz force indicates an occurrence of energy exchange between the flow and the outside. In order to estimate this energy exchange, an electrical efficiency η e defined by the following equation was used. < j E > ηe = (9) < j B u > When the current is alied from the outside, the electrical efficiency becomes negative and it is ositive for MHD ower generation. Combining Eq. (9) with Eq. (5), we can estimate the energy exchange, and the energy addition EA is related to the interaction arameter S and to the electrical efficiency η e. ( κ 1)M EA = 1 Sη e + ( κ 1)M1 (10) Figure 3 shows results of calculation for M 1 =.5. Fig.3-(a) indicates that when the electrical efficiency η e is constant, Mach number M in suersonic solution decreases with the negative interaction arameter. On the other hand, M in subsonic solution increases with the negative S. The increase of Mach number M in subsonic solution suggests the weakening of shock wave and in fact, the static ressure ratio shown in Fig.3-(b) is decreasing with the negative S for constant η e. If we look a change of M along the broken line lotted in Fig.3-(a), we notice that with a decrease of electrical efficiency, M increases. At the same time, the strength of shock wave is weakened (see Fig.3-(b)). Here, it should be recalled that at low electrical efficiency, most of the work done by the flow against the Lorentz force is converted into Joule dissiation. The Joule dissiation results in gas heating. In order to illustrate the gas heating, the energy addition is lotted in Fig.3-(c). The negative energy addition EA denotes an extraction of thermal energy from the flow and it is an enthaly extraction in the MHD ower generation. Along the broken line (S is fixed) shown in Fig.3- (c), the negative energy addition is decreasing with the decrease of electrical efficiency, and this indicates a rise of gas temerature. Therefore, in order to reduce the shock strength, the flow should not be cooled by the enthaly extraction, but be heated. Total ressure ratio is lotted in Fig.3-(d). It should be noted in this figure that the total ressure is lost by the imose of the negative Lorentz force. For examle, the total ressure ratio for S=0 is whereas it is 0.51 at oint P1 (S= and η e =-0.8). This indicates a very Fig.3. Effect of the negative Lorentz force on Mach number (a), static ressure ratio (b), energy addition (c) and total ressure (d) 108

5 imortant fact that the weakening of shock wave by the negative Lorentz force is accomanied by total ressure loss. From Fig.3-(d), it can be seen clearly that the total ressure loss increases with the decrease of electrical efficiency. Therefore, it is said that although the low electrical efficiency or the energy addition is useful for reducing the shock wave strength, a remarkable total ressure loss is induced. Particularly, in the case of energy addition, the energy is injected into the flow through the current sulied from the external ower suly. This accomanies the total ressure loss in an electrical ower generation. Concequently, the total ressure is always lost in the energy injection as a whole. 3.3 Flow Control by Positive Lorentz Force Effect of the ositive Lorentz force (S>0) on the flow is shown here. For this urose, we introduce an acceleration efficiency η ac defined by < j B u > 1 ηac = = < j E > ηe (11) In this case, the energy addition into the flow is exressed by the following equation. ( κ 1)M EA = 1 1 S (1) + ( κ 1)M1 ηac Calculated results are shown in Fig.4-(a) and (b). Suersonic solutions in Fig.4-(a) indicate that for acceleration efficiency of 0.1, 0.3 and 0.5, the suersonic flow is not accelerated but decelerated even if the Lorentz force is acting as flow acceleration. This is because the gas heating which results in the flow deceleration (see Fig.- (b)), is dominant comared with the acceleration due to the Lorentz force. Therefore, η ac must be high for an efficient acceleration. Looking at subsonic solutions, we notice the remarkable flow acceleration for η ac =0.1 and 0.3. In the case of η ac =0.3, the total ressure is not decreased, but increased slightly since the work done by the Lorentz force is added into the flow. Therefore, it seems that the energy addition under the ositive Lorentz force is the best way to reduce the shock wave strength without total ressure loss. But, it requires as mentioned above, the additional energy. 4. Exerimental Observation of Suersonic Flow with and without Shock Wave under Lorentz Force Fig.4. Effect of the ositive Lorentz force on Mach number (a) and total ressure (b) The above analytical results indicate that when S and η e are given, there exist two solutions: the suersonic flow and the subsonic flow with shock wave. In order to know which flow aears in an actual situation, exerimental data obtained at Tokyo Institute of Technology, are reviewed in this section. At T.I.Tech, exerimental studies on the closed cycle MHD ower generation have been carried out using shock-tube driven disk MHD generators. Through the exerimental studies, many hotograhs of the flow under the Lorentz force and the negative energy addition (=enthaly extraction) were taken. Figure 5 shows tyical two hotograhs taken by a high seed-framing camera. Fig.5-(a) shows a hotograh of the flow at low seed fraction (=S.F.) of Although the Faraday current j θ along the azimuthal direction of the disk MHD channel has not be measured in this exeriment, the interaction arameter S = j θ B ρu is regarded small because of the low seed fraction. In this hotograh, we can see many luminous layers at the MHD channel and they extend downstream. These layers are weak comression waves originating from the rear edges of swirl vanes installed in the nozzle. But, it is noted in this hotograh that there is no shock wave 109

6 in the disk MHD channel. This is confirmed also by measured static ressure distribution along the radial direction. (A) S.F.= (B) S.F.= Fig.5. Photograhs of flow taken by high seed camera (B=3.0 Tesla, P 0 =0.3MPa, R= 0.14ohms) On the other hand, we can see a circular bright layer in the hotograh (b) taken at high seed fraction of The measured static ressure distribution indicates that the bright circular layer is a shock wave. In this case, the interaction arameter is regarded large and thus, this shock wave is ascribed to the strong Lorentz force. From many hotograhs and measured static ressure distributions, we can lot a ma of shock formation on the relation between the negative energy addition and the isentroic efficiency. Here, the isentroic efficiency is defined by Eq. (13). EA ηisen = (13) κ 1 P κ 1 0 P01 The result is shown in Figure 6. Solid circles denote the aearance of shock wave and circles denote the shock-free flow. From this figure, it seems that at the same negative energy addition (=enthaly extraction), there are two flow tyes of the suersonic flow and the subsonic flow with shock wave. This oint is discussed later. It is noted in Fig.6 that the isentroic efficiency is ket higher in the shock-free flow for the same negative energy addition. According to Eq. (13), the isentroic efficiency decreases with a decrease in the total ressure ratio of P 0 /P 01. Therefore, the low isentroic efficiency in the flow with shock wave is ascribed to larger total ressure loss due to shock wave. 5. Exlanation of Exerimental Result by the Present Analytical Model Here, we try an exlanation of exerimental results shown in Fig.6. For this urose, the analytical model described in Sec. was used. In order to exlain the exerimental data quantitatively, an area change of the channel should be included in the resent model. This is because the area of the disk MHD channel used in the exeriment, increases along the flow direction and an area ratio (=exit/inlet) is 5.9. But, the objective of the resent simle model is to exlain the exerimental data avoiding a comlex mathematical rocedure. Therefore, the area ratio was assumed 1.0. Furthermore, the inlet Mach number M 1 was assumed.0. In the calculation, the interaction arameter S and the electrical efficiency η e were changed as arameters. Fig.7. Calculated relation between isentroic efficiency and aearance of shock wave Fig.6. Exerimental relation between isentroic efficiency and aearance of shock wave The calculated result is shown in Figure 7. In this figure, the suersonic and the subsonic (shock wave) solutions are lotted by circles and by solid circles, resectively. By comaring Fig.7 with 110

7 Fig.6, we notice a significant difference in values of both the negative energy addition and the isentroic efficiency. And, a larger negative energy addition (=enthaly extraction) is achieved in the exeriments. This is ascribed to the large area ratio in the exeriments. As shown in Fig.3, the resent analytical relations rovide two solutions at the same EA for given S and η e, These two solutions are denoted by A 1 and A in Fig.7. However, in the actual situation, the flows corresonding to these two solutions do not exist simultaneously, but one of them exists. Consequently, it is suggested that S and/or η e were different in the observed two flows. Here, the very imortant and interesting question is still left. Which flow does aear in an actual situation? Concerning this question, the authors do not have a definite answer and therefore this oint should be investigated further. However, the resent calculated results exlain qualitatively the exerimental data of shock formation under the Lorentz force and the negative energy addition. Conclusion The analytical study on characteristics of suersonic flow under the energy addition, the energy extraction and the Lorentz force was carried out. The followings are drawn as conclusions. 1. The simle relations for fluid-dynamical roerties under the energy addition, the energy extraction and the Lorentz force were derived.. The derived relations can exlain the observed henomenon of shock weakening by the heat addition. The heat addition is found also to make the Mach number behind shock wave high. 3. The weakening of shock wave by the negative Lorentz force is always accomanied by the total ressure loss. 4. The simle relations can exlain well the measured relation between the negative energy addition(=enthaly extraction) and the isentroic efficiency of the MHD generator and shock formation. References 1. J.Poggie, Energy Addition for Shockwave Control, AIAA , 30th Plasmadynamics and Lasers Conference, Norfolk, VA, Y.P.Golovachov, et., al, Numerical Simulation of MGD Flows in Suersonic Inlets, AIAA , 31st Plasmadynamics and Lasers Conference, Denver, CO, S.Merriman, Shock Wave Control by Nonequilibrium Plasma in Cold Suersonic Gas Flow, AIAA00-37, Fluids 000, Denver, CO, R.B. Miles, Flow Control by Energy Addition in to High-Seed Air, AIAA00-34, Fluids 000, Denver, CO,