Katı Yakıtlı Roketlerde Daimi Olmayan İç Akışlar Üzerine Bir Sayısal İnceleme

Abstract

Bu çalışmada, çeperlerinden daimi ve üniform kütle girişi olan uzun, dar ve yan kapalı bir silindirin oluşturduğu bir katı yakıtlı roket yanma odası modelinde daimi olmayan akış sayısal olarak incelenmiştir. Korunumlu formda yazılmış süreklilik, momentum ve enerji denklemleri, 2-4, sonlu farklar yöntemi kullanılarak eksenel simetrik bir geometride sayısal olarak çözülmüştür. Çeperlerden yanma odası boşluğuna radyal yönde giren kütle giriş hızına tek ve birden fazla frekansla uygulanan zamana bağlı harmonik değişimler, sistemde meydana gelen rahatsızlıkların kaynağını oluşturmakta ve eksenel yönde zamana bağlı salınımlara neden olmaktadır. Hız değişiminin genliği, Mach sayılarına uygun olarak seçilerek, lineer olmayan etkilerin gözönüne alınabilmesi sağlanmıştır. Akış hızının dönümlü bileşenine dayanarak yapılan incelemeler, silindir çeperlerinin yakınında, daimi akıştakindekine nazaran daha yüksek değerde girdap oluştuğunu ve daha önceki tek boyutlu kararlılık analizlerinde ortaya atılmış olan modellerin aksine, bu girdap alanının ilerleyen akışla beraber silindirin içine büyük oranda yayıldığını ve sonunda tüm geometriyi kapladığını göstermektedir. Bu durum, özellikle bu çalışma kapsamında uygulanan birden fazla uyarıcı frekans modelinin sonuçlarında daha belirgin olarak kendini göstermekte, dönümlü akışın zamana bağlı olarak daha karmaşık bir şekilde ortaya çıkmasına sebep olmaktadır. Girdap alanı incelemeleriyle beraber, toplam hızın eksenel bileşenine göre yanma odası geometrisinin tümünde yapılan çalkantı şiddeti (RMS) incelemeleri, deneysel olarak kurulan modellerden elde edilmiş deneysel ve bunlara dayanarak elde edilen sayısal sonuçlardakilere yakınlık göstermektedir. Ayrıca, hem tek, hem de birden fazla frekansla uyarılan örneklerde, zamana bağlı halde modelin merkezinde hesaplanan RMS değerlerinin, çeperlerden zamana bağlı rahatsızlık verilmeyen deneysel çalışmalardakilerin aksine, daha büyük değerlerde olduğu ortaya konmuştur. Bu sonuç, eksen üzerinde oluşan salınımların kayda değer şekilde ortaya çıkmadığı klasik silindir içi akışlara dayanarak oluşturulmuş bazı türbülans modellerini kullanan sayısal çalışmaların sonuçlarının, katı yakıtlı roket yanma odasında oluşan iç akışın tasvirinde yetersiz kaldığını göstermektedir. Bunlara ek olarak, gözenekli çeperi temsil eden yanal duvarlar yakınında oluşan anlık girdap şiddeti ve bunun sebep olacağı kayma gerilmelerinin zamanla değişiminin, yine aynı bölgelerde oluşan eksenel yönlü statik basınç değişimleriyle beraber değerlendirilmesi sonucunda, bunun, katı yakıtın zamana bağlı olarak erozif ve düzensiz yanmaya uğramasına sebebiyet vereceği söylenebilir. Bu sayısal çalışmada gözlenen erozif yanmaya aday bölgeler, deneysel olarak erozif yanmanın gözlendiği kesitlerle benzerlik göstermektedir.

Unsteady flow dynamics in a model of a solid propellant rocket motor (SRM) is investigated computationally. The model consists of a cylindrical channel with one closed end. Internal shear flow is induced by the mass entry through the porous sidewall (Fig.A.l). Continuity, Momentum and Energy equations in conservative form are non- dimensionalized using the characteristic geometrical and physical magnitudes of the model and numerically solved using the two-four (2-4) finite difference scheme which is a fourth order variant of MacCormack's second order scheme (BAYLISS et al., 1985). Two-four method is known to be highly phase-accurate and therefore very suitable for wave propagation and wave interaction problems in time-dependent flows. The sizes of grids along and across the computational domain are chosen to be equal in the model. Grid point density in the domain is changed for different characteristic Mach numbers in order to capture precisely the gradients in the flow as implied in the asmptotic study of ZHAO et al., (1994) and implemented in the accompanying study by KIRKKÖPRÜ et al., (1996). A steady state solution is required as an initial condition for the unsteady calculations. Boundary conditions for steady flow include an impermeable head-end (solid wall) at x=0, an assumed pressure node at the exit plane x=l (p=l), a specified injection velocity through the porous sidewall which satisfies the blowing condition (v=-l), a no-slip condition for the axial flow speed (u=0) on the sidewall at r=l and symmetry conditions on the centerline r=0 for the axisymmetric cylindrical geometry (Fig. A.2). The analytically calculated velocity profiles derived by CULICK, (1966) for incompressible, rotational, inviscid flow in a long, narrow cylinder are used as starting profiles for the steady, compressible viscous flow computations. By this approach, total computation time required to reach a converged steady flow configuration for specific M and Re numbers is reduced significantly relative to that for a time-marching steady solution from a rest state in the cylinder by initiating wall injection at r=l, t=0. Resulting profiles are indistinguishable from those of Culick, because M=O(10"2-10"1). Once the steady state conditions are obtained, the flow is disturbed by adding an axially distributed unsteady sidewall injection component to the steady value as follows: v(x,r = l,t) = -v0 l + Acos!n- xl(l-cos((Dt))l (1) xvu The amplitude of the unsteady wall injection A is chosen to be A=0(1) in order to include the effect of nonlinear processes (ZHAO et al., 1994 and KIRKKÖPRÜ et al., 1996). In this study A=0.4 is chosen. The other boundary conditions are the same as those of steady state computations. Unsteady computations are carried out for several different axial characteristic Mach numbers (M=0.06 and 0.02) and spatial dependence parameters (n=l, 3, 5, 7). Various single disturbance frequencies (co) are applied to investigate the dynamic response of the flowfield. Afterwards, according to a different approach in this study, multiple frequencies distributed axially along the sidewall of the cylindrical chamber are applied to disturb the steady flowfield. The length of the cylinder is divided into equal subsections and each is disturbed with transient radial mass input of the same form of (1) but with different frequency. Non-resonant frequencies are chosen in order to avoid resonance phenomenon. Investigations about the flow dynamics are based primarily on the unsteady (rotational) part (uv) of the total axial velocity field (u) obtained from uv(x,r,t) = u(x,r,t)-us(x,r)-uP(x,t) (2) where Us is the steady axial velocity and uP is the planar acoustic part, and on the RMS value of the fluctuating part of the velocity defined as = V(^> i = V(u-nr (3) it+T u = -Judt t+T (4) where U is the mean value of the total velocity, u = ü + u'. Performed calculations and their parameters can be seen in Table Al. First eight rows of the table describe the parameters of single frequency disturbing cases whereas the others describe those of multiple frequency cases. Five different disturbing frequencies are randomly chosen. Time dependent values of the static pressure taken in the mid-section of the geometry close to the sidewall (Fig. A. 8- 13), show that with increasing characteristic Mach numbers, the amplitudes of the pressure fluctuations increase. This supports the results of the asymptotic study of ZHAO et al. (1994) which stated that the amplitudes of the fluctuations are of the order of characteristic Mach number (O(M)). No radial pressure gradients in the chamber are detected due to large aspect ratio (length/radius» 1). This is the actual case in solid rocket motors. xvui Vorticity front propagates in short times after the sidewall disturbance is introduced, and eventually fills the entire flowfield, goes to zero on the centerline of the cylinder (Fig A.28-37). When the characteristic Mach number (M) is small, velocity gradients are large (Fig. A.30-32), i.e., vorticities are large, which reflects those found in large scale solid rocket motors. Inversely, when M increases, these gradients decrease. The main reason is that with increasing M, the convective velocity which "carries" the vorticity front towards the centerline, increases. Low frequency disturbing creates large amplitude unsteady vorticity field, especially in the neighborhood of the sidewall. This becomes especially significant if the disturbing frequency is close to the natural frequency of the geometry (Fig. A.32). Large spatial dependence parameter models more realisticly the spatially nonuniform burning rate of the propellant (Fig. A.8-13). Larger n values in (1), cause more complicated rotational flow in the chamber even for single frequency disturbing, Additionally, with increasing n values, amplitudes of the time dependent static pressure decrease. Multiple frequency disturbing results in more complex flow structure compared with that for single frequency calculations, producing more realistic distribution of flow variables. Amplitudes of flow variables decrease with increasing frequencies. The reason for this is that a fluid particle, entering into the chamber, is more frequently affected by different larger frequencies. Time dependent values of the total axial velocity (u) on the centerline (Fig. A.21-27) show that there is significant axial fluctuations even on the centerline due to radial disturbing. Results of RMS calculations based on the axial component of the velocity according to (3), (Fig. A.40-50) are similar to those obtained experimentally (DUNLAP et al., 1974, 1990, YAMADA et al., 1976, TRAINEAU et al., 1986), and computationally (BAUM, 1989, 1990, BAUM and LEVTNE, 1987, SABNIS et al., 1989, LIOU and LIEN, 1995). Generally, with single or multiple frequency disturbing, on the centerline towards the exit, RMS values increase (Fig. A.47-50). This result, which is obtained in this study, appeared because of the sidewall disturbances of 0(1) which give rise to large time dependent fluctuations of axial velocity on the centerline (Fig. A.21-27). At this point, it must be noted that, previous modelling efforts based on the turbulence models may not represent correctly the flow behaviour near the centerline, because they all are developed for steady flows over non-injecting simple geometries. However, in solid rocket motors, time dependent radial velocity disturbances result in axial wave phenomena which, in turn, create axial velocity and pressure fluctuations. This feature must be considered in order to represent the flow behaviour inside solid propellant rocket combustion chambers. xix For all cases, the intensity of the vorticity in other words transient shear stress, field near the sidewall increases towards the exit of the cylindrical chamber (Fig. A. 5 3 -63). Combining these with the results of time dependent static pressure axial gradients close to the sidewall (Fig. A, 8-20), it can be concluded that this may cause scouring of the propellant leading to erosive burning. These results coincide with the previous experimental observations reviewed by KING, (1991).