Proceedings of the 8th International and 47th National Conference on Fluid Mechanics and Fluid Power (FMFP) December 09-11, 2020, IIT Guwahati, Guwahati-781039, Assam, India FMFP2020–094 The Effect of the Forebody Shapes on the Stability of the Axisymmetric Boundary Layer Parth Pandya1, Ramesh Bhoraniya2, Ravi Kant3 1, 2 3 Department of Mechanical Engineering, Marwadi Education Foundations Group of Institutions, Rajkot-360003, India Department of Mechanical Engineering, SOT, Pandit Deendayal Petroleum University, Gandhinagar-382007, India ABSTRACT two steps. In the first step, base flow computation was done, and in the second step, the stability analysis was performed. The base flow computations were done using Finite Volume code ANSYS Fluent. The stability analysis was performed by solving a general eigenvalues problem. Three different axisymmetric forebody shapes named ellipsoid, paraboloid, and sharp-cone with different fineness ratio (FR) of 2.5, 5, and 7.5 were considered. The effects of these forebody shapes have been studied on the axisymmetric and helical modes. This paper represents the effect of the axisymmetric forebody shapes on the stability of an axisymmetric boundary layer developed on a circular cylinder for incompressible flow. We considered sharp-cone, paraboloid, and ellipsoid with fineness ratio (FR) of 2.5, 5, and 7.5. The fineness ratio is defined by the ratio of the length of forebody shapes to the diameter of the cylinder. The incoming base flow velocity is parallel to the cylinder's axis, and hence it is axisymmetric. The parallel base flow is considered, and spatial local stability analysis is performed. The streamwise wave-number () is complex, and wave frequency () is real. The Spectral collocation method was used to discretize the stability equations. The discretized equations, together with the boundary conditions, form a polynomial eigenvalue problem. To understand the effect of forebody shapes and FR, the spatial growth rate (-i) is computed at different streamwise locations at given critical Reynolds number (Re) 12439 and 6070 for axisymmetric and non-axisymmetric modes respectively. It is found that small disturbances are more stable for the sharp-cone and least stable for the ellipsoid. The three different Reynolds number 1000, 4000 and 10000 are considered to understand that for which Re the flow is stable for a given particular shape. 2. LITERATURE REVIEW AND OBJECTIVE Tutty and Price [1] have worked on Boundary layer flow on a long thin cylinder. They studied the boundary layer characteristics and stability in case of the flow over a long cylinder. They found that for the axisymmetric mode (n = 0), the critical Reynolds number is 12439, and for the helical mode (n = 1), it is 1060. Vinod et al. (2002) [2] have reviewed on Stability Analysis of an Axisymmetric Boundary Layer. This research paper shows that transverse curvature has a significant effect on the stability of the boundary layer. They found that for high curvature, the helical mode is dominant and unstable. Vinod and Govindarajan (2006) [3] have worked on Linear and secondary instabilities in incompressible axisymmetric boundary layers with the effect of the transverse curvature. They found that axisymmetric mode becomes unstable above the critical Reynolds number of 12439, and the helical mode is never unstable for curvature above unity. Glauert and Light-hill (1954) [4] have worked on the Axisymmetric boundary layer on a long thin cylinder. This paper represents the investigation of the laminar boundary layer in axial flow about a long thin cylinder. Mérida, Jelliti, and Lili (2016) [5] worked on Laminar instability of parallel and nonparallel flows for the adiabatic flat plate. It represents the effect of Mach No. on the stability of the flow. They found that the critical Reynolds number of nonparallel flow decreasing with increasing Mach No. while the critical Reynolds number of parallel flow increasing with increasing Mach No. Wan, Yang and Zhou and sun (2014) [6] have worked on Linear stability analysis of supersonic axisymmetric jets. This paper represents the study of linear stability of axisymmetric jets with different velocity, momentum thickness, and core temperature. They found that increased velocity and core temperature would increase the disturbance amplification, and hence the flow becomes unstable. Muralidhar and Govindarajan (2015) [7] have worked Keywords: Forebody, Axisymmetric boundary layer, Local stability, spatial stability, Reynolds number 1. INTRODUCTION Hydrodynamic stability is an important theory in fluid dynamics. It includes the stability analysis of the fluid flow and its theoretical approaches. Hydrodynamic stability has significant importance in aerodynamics, marine hydrodynamics, fluid mixing process, and flow-control methods. This paper presents the linear spatial stability analysis of an axisymmetric boundary layer under the effect of forebody shapes on the boundary layer's stability. When fluid flows over the flat surface (i.e., flat plate), it generates the plane boundary layer over the flat surface. Similarly, when fluid flows over the axisymmetric body like a cylinder, it generates the axisymmetric boundary layer over the surface. Most of the underwater vehicles consist of a primary cylindrical surface, which forms the axisymmetric boundary layer when fluid flows over it. The underwater vehicles also have specific forebody shapes that impact the stability of an axisymmetric boundary layer. The present study follows the 1 on Linear stability analysis of the boundary layer over a long cylinder with rotation. This paper represents the stability analysis of a cylinder, which rotates about its axis with axial flow. They found that as the rotation increases, the critical Reynolds number will decrease; hence flow will become unstable. Kumar and Mahesh (2018) [8] have worked on the analysis of the axisymmetric boundary layers. In this article, axisymmetric boundary layers were studied using an integral analysis of the governing equations for axial flow over a circular cylinder. They found that the presence of transverse curvature increases the skin-friction coefficient (Cf). LIU, WANG, and PIAO (2016) [9] have worked on Linear stability analysis of interactions between mixing layer and boundary layer flows. This paper represents the linear stability analysis of incompressible boundary layer flows. Three Mixing layers are mentioned in this paper: - (1) (Wake + Shear) layer + boundary layer (WSBL) (2) Wake layer + Boundary layer (WBL) (3) Shear layer + Boundary layer (SBL). They found that critical Reynolds numbers for the WBL, SBL, & WSBL are 481,492 &, 497 respectively. Xin, Dengbin (2009) [10] have worked on the Nonlinear Stability of Supersonic Nonparallel Boundary Layer Flows. This paper represents a nonlinear study of disturbance waves in supersonic nonparallel boundary layer flows. Johnson, Pinarbasi (2014) [11] have worked on ‘The effect of pressure gradient on boundary layer receptivity. They found that APG leads towards receptivity, which is more than double for the FPG. The present study's main objective is to understand the effect of different forebody shapes like sharp-cone, ellipsoid, and paraboloid on the stability of the axisymmetric boundary layer. The spatial growth rate has been computed for different shapes and FR at different streamwise locations for a given critical Reynolds numbers. In Figure 1, the curve p-a is a circular arc which is an inlet of flow domain and inlet boundary condition is applied to it. The Reynolds number was calculated considering the body radius of the cylinder (a) as a characteristic length. Re = U a (1) Where , U, and are density, free-stream velocity, and dynamic viscosity of the fluid, respectively. The stability analysis was performed considering the parallel flow assumption of modified base flow due to the forebody (bof). The governing stability equations have been derived in the cylindrical polar coordinates (x, r, ). The normal mode form of the disturbances was considered. The perturbations equations can be written as, q( x, r , , t ) = qˆ (r )ei( x+n −t ) , Where, q = [u, v, w, p]. The stability equations were derived using standard procedure and are as follow, u t v t w t u x The enhanced understanding and knowledge will be useful in deciding appropriate flow control strategy and thus to design and operate energy efficient vehicles with reduced emission. +v +U U r v x +U + v r +U =− w x + u x p r =− v r + p =− + x + 1 2 u 2 u 1 u 2 1 u + + + Re x 2 r 2 r r r 2 2 1 2 v 2 v 1 v 1 2 v v 2 w + + + − − Re x 2 r 2 r r r 2 2 r 2 r 2 1 p r 1 w r + (2) (3) 1 2w 2w 1 w 1 2w w 2 v + + + − + (4) Re x 2 r 2 r r r 2 2 r 2 r 2 =0 (5) 3.2 Boundary conditions 3. PROBLEM FORMULATION The appropriate boundary conditions were considered in the radial direction for the numerical solution of the eigenvalue problem. On the surface of cylinder, no-slip and no-penetration conditions were applied. So the magnitude of all velocity perturbations are zero at the wall. U(x, a) = 0, v(x, a) = 0, w(x, a) = 0 3.1 Geometry and Governing stability equations A circular cylinder with different forebody shapes was considered in the axial stream of the incompressible fluid. The streamwise locations between the lines ab and dc were selected for the stability analysis. The boundary layer development starts from the stagnation point O, as shown in Fig.1, and the forebody (bof) significantly affects the boundary layer developed on the main body (cylinder). In the radial direction, far away from the wall it is assumed that all perturbations are going to decay. So at far-field, all velocity and pressure perturbation approaches to zero magnitude. U(x,) = 0, v(x,) = 0, w(x,) = 0, p(x,) = 0 3.3 Discretization The discretization of the stability equations was done using the Chebyshev Spectral Collocation method. The Chebyshev polynomial generates a non-uniform grid and generates more collocation points towards the end. It is a sufficient arrangement for the boundary layer problems. j yc = cos Where, j = 0, 1, 2, 3……….m m Figure 1: schematic diagram of the boundary layer on a circular cylinder with the fore-body 2 (15) Where m is the number of collocation points in wall-normal direction. The following equation applies grid stretching: yr = yi L y (1 − yc ) L y + yc (L y − 2yi ) +a A streamwise location with x = 1.21 m (from the leading edge of a cylinder) and radial location, r = 0.011 m (from the cylindrical surface), were selected to perform this test. Table 1: Grid convergence study for the base flow computations. (16) When all the partial derivatives are discretized by the Chebyshev Spectral Collocation method, the equations (2) – (5) can be written in the matrix form as given below. A11 A21 A31 A41 A12 A22 A32 A42 A13 A23 A33 A43 A14 A24 A34 A44 u v + 1 w p B11 B21 B31 B41 B12 B22 B32 B42 B13 B23 B33 B43 B14 B24 B34 B44 u v w p C11 C12 C13 C14 + 2 C21 C22 C23 C24 = 0 Sr. No Grid size U (m/s) % error V (m/s) % error #1 1001×125 0.0154920 --- 2.5416 --- #2 1415×177 0.0154954 0.02190 2.5344 0.282 #3 2001×251 0.0154962 0.00516 2.5290 0.212 Table 1 shows the grid convergence test for the different grid sizes. The grid size #3 has been selected for the base flow computations in all results presented here. A SIMPLEC algorithm with the second order upwind spatial discretization scheme was used in all computations. (17) C31 C32 C33 C34 C41 C42 C43 C44 4.2 Baseflow profile for the different forebody shapes A + B + 2 C = 0 The baseflow profile is obtained for the different forebody shapes within the boundary layer region to understand the effect of forebody shapes on the profile. We have plotted the graph of axial baseflow velocity ‘U’ versus radius ‘r’ (wall-normal distance or y coordinate) at different streamwise locations. We have considered the two different streamwise locations such that first location is near the leading edge and the second location is far away from the leading edge (or we can say towards down-stream direction) (18) The equation (18) is a polynomial type eigenvalue problem with the order of two. This equation was solved using the ‘ployeig’ MATLAB function. The above equation solution gives complex eigenvalues as complex wave-number ( = r + ii) If i > 0 the flow is stable, i = 0 the flow is neutrally stable, and i < 0 the flow is unstable. Where, i is the spatial growth rate of the disturbance. 4. BASEFLOW SOLUTION 4.1 Governing baseflow equations and Grid convergence study The baseflow solution was obtained using Finite Volume Code, ANSYS Fluent. The steady Navier-Stokes equations were solved in the axisymmetric domain with the three different shapes of the fore bodies; ellipsoid, paraboloid, and sharp-cone. The baseflow computations were done considering the domain p-o-b-c-d-a-p, as shown in Figure 1. The governing baseflow equations are given below: U U x +V U r =− P x + 1 2U 2U 1 U + + Re x 2 r 2 r r V V P 1 2V 1 V 2V V U +V =− + + + − x r r Re r 2 r r x 2 r 2 V r + V r + U x =0 (19) (a) (b) Figure 2: Baseflow velocity profile for different forebody shapes for FR = 2.5 & Re = 1000 at streamwise location (a) x = 0.9 (b) x = 50 (20) The figure 2 (a) and (b) show the effect of forebody shapes on the baseflow profile at the streamwise location x = 0.9 which is near the leading edge and x = 50 which is towards the downstream direction. As shown in figure 2 (a), the forebody shapes have significant effect on baseflow profile near the leading edge but as shown in figure 2 (b), the forebody shapes have negligible effect on the baseflow profile. Hence the effect of forebody shapes on the baseflow profile is significant near the leading edge only. (21) Where U and V are the base velocity components in the axial and radial directions, appropriate boundary conditions were applied at the inlet, outlet, wall, and free-stream for the numerical solution of the N-S equations. Grid convergence tests were done to check the proper grid size and accuracy of the solution. 3 5. CODE VALIDATION N = 0 and N = 2 respectively. The least stable eigenvalues are marked by a square in both the eigenspectrum. The least stable eigenvalues marked by square are C = 0.320 + 0i and C =.0.425 + 0i are in close agreement with the results of Tutty et al. (2002). The spatial stability analysis has been performed as shown by Eq. 18 to understand the effect of different shapes of fore-body geometry on the stability of axisymmetric boundary layer. To validate the stability computations, a blunt cylinder was considered and temporal and spatial stability results were validated against Tutty et al. (2002). First we performed temporal stability analysis for axisymmetric (N = 0) and nonaxisymmetric mode (N = 2). The critical Reynolds number of 12439 (for N = 0) and 6070 (for N = 2) were considered [1]. As we know that in temporal stability analysis, streamwise wavenumber is real and known while complex wave frequencies are computed as eigenvalues. However, in the spatial stability analysis, wave-frequencies are real and known, and complex wave-numbers are computed as eigenvalues. Table 2: Results of temporal stability analysis of the axisymmetric boundary layer [1]. Input parameters for temporal stability Figure 5: Eigenspectrum for N = 0, Re = 12439, streamwise location x = 47, and Cr = 0.317. The complex eigenvalue marked by square is α = 2.71 + 0.0i. Results N Re Xc c Cr 0 12439 47 2.73 0.317 2 6070 91.1 0.775 0.422 Figure 6: Eigenspectrum for N = 2, Re=6070, streamwise location x = 91.1, and Cr = 0.422. The complex eigenvalue marked by square is α = 0.77 + 0.0i. Figure 3: Eigenspectrum for N = 0, Re = 12439, streamwise location x = 47, and = 2.73. The complex eigenvalue marked by square is C = 0.32 + 0.0i In the second step, spatial stability analysis was performed with obtained complex frequencies from the temporal stability analysis as input (for critical Reynolds number imaginary part of the eigenvalue is zero), and complex wave-numbers are computed as eigenvalues. Figures 5 and 6 present the eigenspectrum of complex wave-number resulting from the spatial stability analysis. The wave-numbers (complex) in Figures 5 and 6 are marked with a square. They have the almost same magnitudes as 2.71 for N = 0 and 0.77 for N = 2 which are in close agreement with the results of Tutty et al. (2002). 6. RESULTS & DISCUSSION The axisymmetric boundary layer’s spatial stability analysis has been performed on the modified baseflow due to the axisymmetric forebodies mounted at the leading edge of the circular cylinder. The ellipsoid, paraboloid, and sharp-cone with FR = 2.5, 5, and 7.5 were considered at Re = 1000, 4000, and 10000. The effect of fore-body shapes has been studied on axisymmetric (N = 0) and non-axisymmetric modes (N = 2). The grid convergence test was also performed for the stability computations. The stability computations have been done for the domain of cylinder only. Figure 4: Eigenspectrum for N=2, Re = 6070, streamwise location x = 91.1, and = 0.775. The complex eigenvalue marked by square is c= 0.425 + 0.0i In the first step, a temporal stability analysis was performed for N = 0, Re = 12439, streamwise location x = 47, and = 2.73 and then for N = 2, Re = 6070, streamwise location x = 91.1, and = 0.775. Figure 3 and 4 show the eigenspectrum for the 4 6.1 Axisymmetric mode (N = 0) favourable pressure gradient makes the boundary layer stable. Figure 8 also indicates that the effect of FR on the spatial growth rate is significant near the boundary layer's leading edge while it gradually reduces in the streamwise direction towards the downstream. Figure 7: Variation of the spatial growth rate (-αi) for Re = 12439, FR = 2.5, N = 0 for different axisymmetric forebody shapes. Figure 7 shows the variation of the spatial growth rate (-αi) of the perturbations at the different streamwise locations for axisymmetric mode, Re = 12439, and FR = 2.5. The comparison shows that the spatial growth rate of the disturbances is highest for the elliptical forebody and the least for the sharp cone towards the leading edge. This indicates that the boundary layer with the sharp-cone forebody is more stable, and the boundary layer with the ellipsoid is less stable. At near the critical location x = 47, the spatial growth rate is almost zero for all the three forebody shapes. In the streamwise direction ahead of the critical position, again, spatial growth starts to reduce, which indicates that the boundary layer is going to relaminarize again. The important observation is that the difference in the spatial growth rate at different streamwise locations has been reduced between all the three forebody shapes. It proves that the effect of the forebody shapes on the stability of the boundary layer is dominant near the leading edge only. The effect of these shapes is almost the same away from the leading edge or towards the downstream. Figure 9: Variation of the spatial growth rate (-αi) in the streamwise direction for paraboloid forebody for N = 0, FR = 5, and ωr = 0.865 for different Re. Figure 9 presents the spatial growth rate of the disturbances for paraboloid forebody with FR = 5 and forcing frequency ωr = 0.865. The diagram shows that the spatial growth rate at any particular streamwise location is highest for the Re=10000 and lowest for the Re = 1000. It proves that increased Reynolds number makes the flow locally unstable, and at critical Reynolds number, it becomes fully unstable, and transition to turbulence occurs. 6.2 Non-axisymmetric (Helical) mode Azimuthal wave number N = 2 was selected at Critical Reynolds number with FR = 2.5 to understand the effect of forebody shapes on the axisymmetric boundary layer's helical modes. Figure 10 shows the comparison of the spatial growth rate for different forebody shapes at Re = 6070. The behaviour for the helical mode N = 2 has been found qualitatively similar to the axisymmetric mode. Figure 8: Variation of the spatial growth rate (-αi) for ellipsoid in the streamwise direction for N = 0, Re = 12439 for different FR. Figure 8 shows the effect of different fineness ratio on the spatial growth rate of the disturbances for ellipsoid forebody in the streamwise direction. For a given forebody shape (ellipsoid), the boundary layer is more stable with FR = 7.5 and less stable for FR = 2.5. It evident that increased FR of the forebody reduces the spatial growth rate of the disturbances, and the boundary layer becomes more stable. The increased FR increases the sharpness of the forebody, which develops a more favourable pressure gradient. It is well established that a Figure 10: Variation of the spatial growth rate (-αi) in the streamwise direction for N = 2, Re = 6070, and FR = 2.5 for different forebody shapes. The x = 91.1 location is the critical point location for the helical mode N = 2. It has been observed here that the spatial growth rate is highest for the ellipsoid and the lowest for the sharp cone. Similarly, the spatial growth rate difference is more significant 5 at the leading edge of the boundary layer and gradually reduces in the streamwise direction. Beyond the critical point, again, spatial growth rate reduces, thus boundary layer relaminarize again towards the downstream. The spatial growth rate for the ellipsoid forebody was highest, while that of sharp-cone was found to be the least one for a given Reynolds number and FR. The forebody with FR = 7.5 has the least spatial growth rate, while a forebody with FR = 2.5 has the highest spatial growth rate. The perturbation was found spatially growing until the critical location and decayed when Reynolds number smaller or equal to Critical values. The perturbations were continuously growing in the streamwise direction for Reynolds number larger than Critical value. NOMENCLATURE Re α ρ ω N U, V, W Figure 11: Variation of the spatial growth rate (-αi) in the streamwise direction for ellipsoid for N = 2, Re = 6070, for different FR Figure 11 shows the variation of the ellipsoid spatial growth rate at Re = 6070 and N = 2. The spatial growth rate is found the least for FR = 7.5. u, v, w U a Reynolds number Viscosity of water Streamwise wave-number Density of water Wave frequency Azimuthal wave-number Baseflow velocity in x, r and directions respectively Disturbance velocity in x, r and directions respectively Free stream velocity Body radius of a cylinder [m2] Ns/m2 1/m [kg/m3] [rad/s] 1/m m/s m/s m/s m REFERENCES [1] O. R. Tutty and W. G. Price, Boundary layer flow on a long thin cylinder, Physics of Fluid (2002) [2] N. Vinod, H. Balakrishnan and R. Govindarajan, Stability Analysis of an Axisymmetric Boundary Layer, Asian Congress of Fluid Mechanics (2002) [3] N. Vinod and R. Govindarajan, Linear and secondary instabilities in incompressible axisymmetric boundary layers: Effect of transverse curvature, J. Fluid Mech. (2006) [4] M. B. Glauert and M. J. Lighthill, the Axisymmetric Boundary Layer on a Long Thin Cylinder, the Royal Society (1954) [5] Y. Mérida, M. Jelliti and T. Lili, Laminar Instability of Parallel and Nonparallel Flows of Adiabatic Flat Plate, Journal of Applied Fluid Mechanics (2016) [6] Z. Wan, H. Yang, L. Zhou, and D. Sun, Linear stability analysis of supersonic axisymmetric jets, Theoretical and Applied Mechanics Letters (2014) [7] S. Muralidhar and R. Govindarajan, Linear stability analysis of boundary layer over a long cylinder with rotation, Conference: Congress Francais de Mecanique (2015) [8] P. Kumar and K. Mahesh, Analysis of axisymmetric boundary layers, Journal of Fluid Mechanics (2018) [9] F. Liu, Y. WANG and Y. Piao, Linear stability analysis of interactions between mixing layer and boundary layer flows, Chinese Journal of Aeronautic (2016) [10] G. Xin and T. Dengbin, Nonlinear Stability of Supersonic Nonparallel Boundary Layer Flows, Chinese Journal of Aeronautic (2009) [11] M. Johnson and A. Pinarbasi, The Effect of Pressure Gradient on Boundary Layer Receptivity, Flow Turbulence and combustion (2014) Figure 12: Variation of the spatial growth rate (-αi) in the streamwise direction for paraboloid forebody shape for N = 2, FR = 5, and ωr = 0.327 for different Re Figure 12 shows the spatial growth rate of the disturbances for different Reynolds numbers at forcing frequency of ωr = 0.327. Re = 10000 is above the Critical value of the Reynolds number. Thus, spatially flow is found unstable as well as in the streamwise direction. Also, spatial growth increases towards the downstream. It shows that the flow is unstable. 7. CONCLUSIONS Local spatial stability analysis of the axisymmetric boundary layer has been performed to study the effect of axisymmetric forebody shapes on the stability of the boundary layer. We considered ellipsoid, paraboloid, and sharp-cone geometry at Re = 1000, 4000, and 10000 and spatial growth rate of the perturbation was computed at different streamwise locations. The polynomial eigenvalue problem was solved using the ployeig function of the MATLAB. It has been found that forebody shapes have a significant effect on the base flow and stability characteristic near the leading edge of the boundary layer. Towards the downstream sufficiently away from the leading edge (x = 50), the spatial growth rate for all three forebody shapes found almost the same. 6