GT2017-64451

EFFECT OF HEAT TRANSFER ON PIPE FLOW STABILITY Ce Zhang School of Aeronautics and Astronautics Shanghai Jiaotong University Shanghai China zhangce Wei Ma School of Aeronautics and Astronautics Shanghai Jiaotong University Shanghai China mawei Wensheng Yu School of Aeronautics and Astronautics Shanghai Jiaotong University Shanghai China yuws Jinfang Teng School of Aeronautics and Astronautics Shanghai Jiaotong University Shanghai China tjf ABSTRACT The compressibility of flow field has an important effect on flow stability However when the compressibility is considered the effect of Mach number is often considered while the effect of heat transfer is always neglected in the existing flow stability studies Linear stability analysis tools based on compressible Orr Sommerfeld O S equations and linearized Navier Stokes equations in cylindrical coordinate system are established in this paper These equations are numerically solved by using Chebyshev spectral collocation method and pseudo modes are eliminated Linear stability analysis of pipe flow with heat transfer whose average flow field is obtained by CFD simulation is carried out The results show that for spatial modes the heating effect of the wall makes pipe flow more unstable while cooling effect of the wall makes pipe flow more stable For global modes of pipe flow the frequency of global mode decreases when the wall cools the flow and the decrease of mean temperature of pipe flow leads to the improvement of global mode stability INTRODUCTION The linear stability theory of flow field can analyze the growth or decay of small perturbation in the flow field and it is originally used to predict the transition and instability of laminar flow 1 2 Since the fluctuation in the fully developed turbulent flow field can be regarded as a small perturbation relative to the time averaged flow field the linear stability analysis of the turbulent flow can predict the development of the fluctuation in the flow field and predict some important turbulence flow structures In recent years flow field stability analysis has been used to predict the coherent structure of turbulent flow 1 3 4 and the self oscillation characteristics of flow field 1 3 5 8 so it plays an important role in predicting the stall of compressor 5 instability of the combustion chamber 3 6 7 instability of turbine tip clearance leakage vortex 8 and other areas of research In the case of a cold turbulent flow not involving heat transfer the results of the stability theory are in good agreement with the numerical simulations and experimental results but for the hot swirl with flames there are significant differences between the results of linear stability analysis based on incompressible O S equations and the conclusion of combustion experiments 6 7 and the experiments found that different types of flame have different effects on the self oscillation characteristics of flow field 7 This indicates that heat transfer plays an important role in the flow stability and the linear stability of flow with heat transfer should be analyzed based on compressible linear stability equations There are a lot of heat transfer processes in the flows of various parts of the aero engine and it is important to study the influence of heat transfer on the stability of aero engine s inner flow field The flow field in the aero engine is very complicated and involves many physical processes Therefore most studies have studied the influence of heat transfer on the flow stability of the simple basic flow field It has been shown that the effect of heat transfer on the flow stability is related to the flow field type For the plate boundary layer Guo 9 studied the parabolic stability equation and found that wall cooling Proceedings of ASME Turbo Expo 2017 Turbomachinery Technical Conference and Exposition GT2017 June 26 30 2017 Charlotte NC USA GT2017 64451 1Copyright 2017 ASME stabilized the boundary layer flow For the free jet Monkewitz and Sohn 10 analyzed the absolute instability of the flow by numerically solving the linearized compressible Euler equations and the results show that it would be easier to stimulate the absolute unstable mode if the temperature is higher than surrounding environment For the flow field with non uniform density Fung and Kurzweg 11 analyzed the stability criterion of the linear Euler equation by analytic method and found that the negative radial gradient of the density of the jet flow make flow unstable which is consistent with the mechanism of centrifugal stability For the hot swirling flow the experimental study of swirl combustion 7 shows that different types of flame have different effects on the swirl stability V shaped flame makes the flow more stable and the annular flame makes the flow more unstable However for the pipe flow although the influence of perturbation amplitude and fluid type on its stability have been studied 12 13 the pattern and mechanism of heat transfer s effect on flow stability of pipe flow are not fully understood In summary heat transfer has an important influence on the stability of the flow but there is no general conclusion about the mechanism of its influence and it has not been studied sufficiently for the basic pipe flow In this paper a compressible linear stability analysis tool is developed to analyze the effect of heat transfer on the flow stability of a round pipe under turbulent conditions and the physical mechanism is discussed LINEAR STABILITY ANALYSIS When the linear stability of turbulent flow field is analyzed the fluctuation of the flow field is regarded as a small perturbation relative to the time averaged flow field and the development of the perturbation is calculated by the linear stability equation Therefore it is necessary to study the time averaged flow field and the linear stability equation O S equations for the linear stability of pipe flow In this paper the CFD simulation is used to obtain the time averaged flow field of a pipe The Chebyshev spectral collocation method is used to solve the compressible O S equations numerically 1 CFD simulation of the pipe flow The research object is a pipe whose radius R 0 077 m length L 1 54 m length and radius ratio L R 20 so the pipe has a sufficient length for the pipe flow being fully developed ICEM software is used to generate structured grids for the pipe The number of meshes is about 1 million the y value is less than 1 and the mesh quality is above 0 75 which satisfies the requirements of CFD simulation In this paper the pipe flows of Re 900 and Re 9000 are studied respectively The corresponding inlet mass flows are 5 10 4 kg s and 5 10 3 kg s respectively and the outlet static pressure is 1 atm Both cases are treated as turbulent flow The boundary condition for wall is no slip wall In order to consider the heat transfer effect in the simulation a set of different wall temperatures and incoming flow temperatures are given as boundary conditions as listed in Table 1 Using this set of boundary conditions we can investigate the effect of various inlet temperatures and wall temperatures on the pipe flow stability ANSYS FLUENT 14 5 is used to simulate the pipe flow The simulation method is RANS steady simulation We select the S A turbulence model as turbulence model and the SIMPLE algorithm as pressure and velocity coupling algorithm In order to ensure that the compressibility of the gas under the heat transfer can be correctly simulated the gas model in the simulation is compressible ideal gas Figure 1 shows an example of the axial velocity contour As can be seen from the figure after the transition of the inlet section the flow in the pipe is fully developed both thermally and aerodynamically Table 1 Temperature boundary condition of pipe flow Temperature of wall K 300 500 700 900 1100 Temperature of inlet flow K 300 500 700 900 1100 Fig 1 Axial velocity contour of typical pipe flow obtained by CFD Re 9000 2Copyright 2017 ASME 2 O S equations For the quasi parallel flow in the cylindrical coordinate system the linear stability theory supposes that the perturbation in the flow field can be decomposed into a series of eigenmodes and the characteristic perturbation mode is assumed to have the following form iz mt u rz tF r e 1 In this formula r z t are dimensionless radial circumferential axial coordinates and time respectively m are dimensionless axial eigenvalue circumferential eigenvalue and temporal eigenvalue respectively m should be an integer while should be complex numbers F r is the shape function of the disturbance mode and F rcan be seen as the distribution function of wave amplitude along the radius The O S equations can be obtained by substituting the expression of the characteristic disturbance mode into the linearized N S equation If a linearized N S equation with no energy equation is substituted then the incompressible O S equations will be obtained 14 If the linearized N S equation contains the energy equation i e the complete N S equation then after substituting the characteristic modal form the compressible O S equations 15 will be obtained According to the dispersion relation the linear stability analysis of flow field can be divided into temporal analysis spatial analysis and spatial temporal analysis In spatial analysis the spatial eigenvalues will be calculated while m are known In the temporal analysis the temporal eigenvalues will be calculated while m are known In the space temporal analysis both and are unknown For turbulent flow the spatial mode is equivalent to solving the flow field fluctuation structure along the streamline As the prediction of the structure has obvious physical significance this paper will use spatial analysis to solve compressible O S equations under the cylinder coordinate In this paper Chebyshev spectral collocation method 14 is used to solve the O S equations numerically This method is accurate fast and easy to program The principle of the Chebyshev spectral collocation method is transforming O S equation into a matrix and solving the eigenvalue of the matrix Therefore the number of eigenvalues and the number of eigenmodes are the same as the order of the matrix which contains a large number of pseudo modals Most of the studies eliminate pseudo modes by experience 3 9 and the removal methods used in these papers lack versatility According to the mathematical principle of Chebyshev spectral collocation method and the related physical laws of flow stability we find that the real modal has the following three characteristics I With the increase of the number of collocation points the convergence rate of each mode obtained by Chebyshev spectral collocation method are different and the convergence speed of eigenvalues corresponding to the real modes is faster II Boundary conditions have the following effects on the perturbation modes far field boundary conditions make the perturbation decays rapidly when the radius is large The non slip boundary condition at the wall will suppress the perturbation growth near the wall Therefore the amplitude function F r of the real mode decays rapidly when the radius r is large and the maximum amplitude point is closer to the axis of the flow field whether free jet or pipe flow 3 III According to the turbulence energy cascade theory 16 in the turbulent flow the small scale turbulent structure has high dissipation and short lifespan while the large scale pulsation structure has high energy and long lifespan Therefore the spectrum of the amplitude function of the real mode in turbulence is dominated by the low frequency component and there is almost no high frequency component Among the above three characteristics I is used for most studies using the Chebyshev spectral collocation method Oberleithner et al 3 used I II to pick the real mode while III is only used in this paper This paper synthesizes I II and III and obtains a set of mode selection method with high universality and high accuracy The Chebyshev spectral collocation method and the mode selection method are programmed as MATLAB programs to solve the compressible O S equations In order to verify the correctness of the program the linear stability of the two classical quasi parallel flows Poiseuille flow 14 and Batchelor vortex 17 is analyzed and the eigenvalues calculated by MATLAB are compared with those in literatures 14 17 for the comparison of the eigenvalues as Table 2 lists We can see that the MATLAB code used in this paper can accurately analyze the linear stability of the flow Table 2 Comparison of the eigenvalue obtained by present code with literature Flow Type Poiseuille Flow m 0 14 Poiseuille Flow m 1 14 Batchelor Vortex 17 Present code 0 5199883085 0 0208406011i 0 5352510830 0 0172276439i 0 36517628735 0 2229717124i Literature 14 17 0 51999 0 020836i 0 53525 0 017228i 0 3651762873 0 2229717124i 3Copyright 2017 ASME 3 Linearized N S equations The global modes of the flow stability are related to the global information of flow field and the boundary conditions at inlet and outlet Therefore the O S equations are not the best choice to analyze the global mode of the flow stability as they were derived under the quasi parallel flow assumption In this section we propose a linear eigenvalue equation set based on linearized N S equations which can be used to compute the global modes of flow instability for both quasi parallel flow and non parallel axisymmetric flow According to the linear stability theory the flow field is assumed to consist of mean flow and small disturbance rrr zzz VVV VVV VVV ppp 2 where the over bar represents the mean flow and over bar represents the small disturbance If the mean flow is axisymmetric then the disturbance term can be assumed to be in the form of harmonic decomposition i mt i mt rr i mt i mt zz i mt r z e VV r z e VVr z e VV r z e pp r z e 3 where the over bar represents the amplitude of disturbance term After substituting the harmonic decomposition into the 3D linearized N S equations we can formulate an eigenvalue equation as follows 22 22 2 2 rrrzzz T rz CCDDimF rzrz m FGimJimKLqi H rzr z qV V Vp 4 where rrrzzz C CD DF FG H J K L are coefficient matrixes determined by mean flow field Both compressibility and viscosity is considered in this equation set is the temporal eigenvalue of global mode The real part of represents the frequency of global mode The imaginary part of represents whether the global mode is stable with negative value or unstable with positive value The numerical discretization of eq 4 can be performed by applying 2D Chebyshev spectral collocation method We will use equation set 4 to analyze the absolute instability of pipe flow The boundary conditions of perturbations waves at inlet and outlet are treated as 1D non reflecting boundary 18 RESULTS AND DISCUSSION 1 Spatial mode O S equations require the flow to be quasi parallel flow the time averaged flow field must be axisymmetric and constant along the axis the flow parameters are only varied along the radius The time averaged flow field of the pipe flow is axisymmetric flow but it changes along the axial direction In order to apply O S equations to the pipe flow the technique called local stability analysis 3 is used in this paper a series of cross sections perpendicular to the axial direction are evenly distributed from the pipe inlet to the pipe outlet The flow field of each section can be regarded as a quasi parallel flow field and then the linear stability analysis of the flow field in these cross sections is carried out and the eigenvalues of each section are obtained by using linear stability analysis In the linear stability analysis the imaginary part of the axial eigenvalue i represents the growth rate of the perturbation in the axial direction when the imaginary part is negative the perturbation will increase and if the imaginary part is positive the disturbance will decay The axial eigenvalues of each section are calculated and the distribution of the growth rate along the axial coordinate can be obtained The time averaged flow field data are post processed and the distribution of the parameters in different cross sections along the radius is obtained The O S equations are solved by Chebyshev spectral collocation method The temporal eigenvalue is 0 5 and the imaginary part of the spatial eigenvalue i is calculated on 14 cross sections evenly distributed from the pipe inlet to the pipe outlet along the axial direction The results of CFD simulation and linear stability analysis for the case where Re 900 are shown in Figure 2 and Figure 3 In Figure 2 the inlet temperature is 700 K and the wall temperature is 300 K blue line 700 K black line and 1100 K red line respectively A cross section near the inlet a cross section in the middle and a cross section near outlet are chosen as the typical cross sections the position of the typical cross section is indicated by a vertical line in Figure 2 and Figure 3 and its velocity temperature and density along the radius are plotted at Figure 2 b and Figure 3 b It can be seen that either increasing or decreasing the wall temperature i remains positive and increases with the increase of the axial coordinate z the growth rate of i decreases gradually in the downstream of the pipe flow which means that the disturbance decays when the disturbance propagates downstream The flow velocity profiles are developed into a 4Copyright 2017 ASME parabolic type and with the development of flow temperature and density profiles gradually become fuller By comparing the stability of each flow in Figure 2 a it can be found that when compared with the case without heat transfer the flow and wall temperature are both 700 K corresponding to the black line in the figure the perturbation decays faster for the case whose wall temperature is 300 K which means the flow whose wall temperature is 300 K is more stable and it can also be found that the flow is more unstable when wall tempera