Direct computation of perturbed impinging hot jets

Computers & Fluids 36 (2007) 259–272 www.elsevier.com/locate/compfluid

Direct computation of perturbed impinging hot jets X. Jiang a

a,*

, H. Zhao a, K.H. Luo

b

Mechanical Engineering, School of Engineering and Design, Brunel University, Uxbridge UB8 3PH, UK b School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK Received 5 July 2005; received in revised form 20 December 2005; accepted 7 January 2006 Available online 22 March 2006

Abstract The unsteady flow and temperature fields of an impinging hot jet at a Reynolds number of 1000 and a nozzle-to-plate distance of 6 jet diameters have been obtained by direct numerical solution of the compressible time-dependent three-dimensional Navier–Stokes equations using highly accurate numerical methods. Effects of an external perturbation on the flow and heat transfer characteristics of the transitional impinging jet have been examined. Oscillatory behaviour induced by the external perturbation has been observed for the impinging jet. The external perturbation leads to the large-scale vortical structures in the primary jet stream, which subsequently lead to the strong oscillatory behaviour of the impinging jet. The vortical structures lead to flow transitional behaviour that enhances mixing of the hot jet with the ambient fluid. It has been observed that the wall boundary layer of the impinging jet is thin. Both the instantaneous and time-averaged wall shear and normal stresses and Nusselt number are examined. Although the external perturbation strongly affects the flow structures in the primary jet stream, it does not have significant effects on the wall stresses and heat transfer characteristics of the impinging jet due to the re-laminarization effect of the wall.  2006 Elsevier Ltd. All rights reserved.

1. Introduction Near-wall fluid flow and heat transfer phenomena are encountered in a broad range of practical applications. In many cases, control of the level of heat fluxes to walls is of crucial importance. The presence of walls also influences the flow dynamics in a significant manner. In these contexts, the study of impinging jets is of particular interest. In addition to a broad range of engineering applications such as cooling and drying, impinging jets are of great value in fundamental academic studies. The impinging flow configuration is of simple geometry but covers a broad range of important flow phenomena, such as large-scale structures, wall boundary layer with stagnation, large curvature involving strong shear and normal stresses, wall heat transfer and small-scale turbulent mixing. As an important ‘‘building block’’ of complex practical applications, impinging jets have attracted much research *

Corresponding author. Tel.: +44 1895 266685; fax: +44 1895 256392. E-mail address: [email protected] (X. Jiang).

0045-7930/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2006.01.015

interest. There have been a substantial number of advanced computational fluid dynamics (CFD) studies using large eddy simulation (LES) and direct numerical simulation (DNS) of impinging jets recently, e.g. [1,3–5,8–10,21,23, 26,31,33], in addition to the extensive experimental studies in this field over the past few decades. However, impinging jets are still poorly understood due to the complex nature of the problem. The near-wall flow field contains a broad range of length scales, ranging from the large-scale vortical structure to very thin thermal boundary layers. The nearwall flow and heat transfer processes are also highly unsteady, which require advanced numerical and experimental techniques to resolve. Therefore, impinging jets deserve more research efforts from both fundamental and application points of view. Due to the rich flow phenomena involved and the geometric simplicity, the impinging flow configuration is ideal for development and validation of near-wall turbulence and heat transfer models [6,7,24]. Moreover, some effective flow control techniques may be tested and developed using the impinging flow configuration. For example, acoustic excitation significantly affects

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the dynamics of impinging jets [1,11,18,30,31], which can be utilized to develop techniques for the control of nearwall fluid flow and heat transfer. Among the various approaches to study impinging jets, the traditional CFD based on the Reynolds-averaged Navier–Stokes (RANS) approach is not suitable for highly unsteady flow phenomena such as impinging jets, due to its intrinsic averaging effect. In experimental studies, it is difficult to isolate and fully understand the individual effects in a coupled, non-linear flow phenomenon. In contrast, advanced numerical approaches such as DNS and LES can provide temporally and spatially resolved solutions. DNS, in particular, resolves directly all the relevant time and length scales, and therefore eliminates the uncertainty introduced by turbulence modelling. The DNS approach has been developing rapidly in recent years due to the significant advancement in both computer hardware and numerical methods. However, application of DNS is limited due to its high computational cost associated with the requirements for extremely fine grids and high accuracy. DNS studies on impinging jets have thus been very few. Some existing DNS studies [10,26] were focused only on the mean flow variables and turbulent kinetic energy budget, but not the vortical structures that are important to the flow dynamics and mixing characteristics. In the meantime, two-dimensional simulations [4,5] are not ideal for transitional or fully turbulent flow because there is a lack of vortex stretching. LES has relatively low computational cost [1,3,8,9,21,23,31,33], but it has not been fully developed, due mainly to the difficulty of constructing robust sub-grid scale models. The study was intended to obtain better understanding on the unsteady dynamics of heated impinging round jet, which has wide industrial applications. Since DNS brings information that is not available by any other means, direct simulations of impinging jets were adopted. The main objective of this study was to examine the effects of an external perturbation on the dynamics of the impinging jets. Therefore a comparative study of the flow characteristics of unperturbed and perturbed impinging jets has been carried out. Effects of perturbation amplitude on the unsteady dynamics of impinging jet have been examined. Unlike the previous experimental study and numerical study based on the traditional RANS approach, the direct simulation approach is able to capture the detailed threedimensional (3D) vortical structures and to identify the effects of an external perturbation on the unsteady dynamics of impinging jets. DNS of impinging jets at different upstream external perturbations was performed in this study. The timedependent simulations aimed to characterize the unsteady flow and temperature fields of the impinging jet prior to further efforts on developing flow control techniques. By performing both spatially and temporally resolved 3D DNS, detailed information on the large-scale structures and unsteady characteristics of the impinging jets have been obtained. The unsteady flow and temperature charac-

teristics and time-averaged flow and temperature fields of the impinging jet under external perturbations of different amplitudes have been investigated. The near-wall heat transfer characteristics, and normal and shear stresses have also been examined. The rest of the paper is organized as follows. Section 2 first presents the governing equations for the flow field, and then introduces the numerical methods and solution conditions for the spatial DNS of the impinging hot jet. Section 3 is a discussion of the numerical simulation results from a comparative study of unperturbed/perturbed impinging jets. The analysis of the results is based on both instantaneous and time-averaged flow quantities. Finally, conclusions are drawn in Section 4. 2. Mathematical formulation The physical problem considered is a heated round jet issuing into an open boundary domain that impinges on a flat surface. It is a free jet with two confinement plates, one confinement plate is the jet nozzle inlet plane and the other one is the impingement plate or the wall, as schematically shown in Reference [5] but with a 3D configuration. Although a temporal DNS with periodic boundary conditions using spectral methods can be performed at a relatively low computational cost [20], they are not suitable for impinging jets with realistic boundaries. A time-accurate spatial DNS provides the best means to gain insight into the complex physics involved in impinging jets. In this study, a spatial DNS based on high-order finite-difference numerical schemes and high-fidelity boundary conditions has been performed for a heated impinging jet, using a recently developed code for spatial DNS of non-reacting and reacting jets [13–16]. The mathematical formulation of the physical problem includes the governing equations, boundary conditions and numerical methods for discretization and solution, as described below. 2.1. Governing equations The flow field is described with the compressible timedependent Navier–Stokes equations in the Cartesian coordinate system (x, y, z), where the z-axis is along the streamwise direction for the head-on impingement and the x–0–y plane is the jet nozzle exit plane that is the domain inlet. In this work, the non-dimensional form of the governing equations has been employed. Major reference quantities used in the normalization are the centerline streamwise mean velocity at the jet nozzle exit (domain inlet), jet nozzle diameter, and the ambient temperature, density and viscosity. To allow sufficient resolution, the impinging jet examined has a relatively low Reynolds number of Re = 1000, at which the impinging flow may display transitional behaviour [32]. The non-dimensional conservation laws for mass, momentum and energy can be written in the following vector form: oU oE oF oG þ þ þ ¼ 0; ot ox oy oz

ð1Þ

X. Jiang et al. / Computers & Fluids 36 (2007) 259–272

where the vectors U, E, F and G are defined as 1 0 q C B B qu C C B C U¼B B qv C; C B @ qw A ET 3 2 qu 7 6 qu2 þ p  sxx 7 6 7 6 7; quv  s E¼6 xy 7 6 7 6 quw  sxz 5 4 ðET þ pÞu þ qx  usxx  vsxy  wsxz 2 3 qv 6 7 quv  sxy 6 7 F¼6 7 4 5 qv2 þ p  syy

p¼

ð2Þ

ð3Þ

ð4Þ

ðET þ pÞv þ qy  usxy  vsyy  wsyz and

2

6 6 6 G¼6 6 6 4

3

qw quw  sxz qvw  syz qw2 þ p  szz

7 7 7 7. 7 7 5

ð5Þ

with e standing for the internal energy per unit mass, u, v, and w for velocity components in the x, y, and z directions, and q for density. For the other variables, p stands for pressure, t for time, the heat flux components are l ðc  1ÞM 2 Pr Re l qz ¼  ðc  1ÞM 2 Pr Re

oT ; ox oT oz

qy ¼ 

l oT ; ðc  1ÞM 2 Pr Re oy

with T representing temperature, and the constitutive relations for viscous stress components are   2 l ou ov ow sxx ¼  2 þ þ ; 3 Re ox oy oz   2 l ou ov ow 2 þ ; syy ¼  3 Re ox oy oz   2 l ou ov ow þ 2 ; szz ¼  3 Re ox oy oz     l ov ou l ow ou þ þ ; sxz ¼ ; sxy ¼ Re ox oy Re ox oz   l ow ov þ ; syz ¼ Re oy oz where M, Pr, and Re represent Mach number, Prandtl number, and Reynolds number, respectively. c is the ratio of specific heats and l the dynamic viscosity. The flow field governing equations are supplemented by the ideal-gas law, which is given by

ð6Þ

2.2. Numerical methods and solution details The solution methods of the governing equations include the high-order numerical schemes for spatial discretization and time advancement. The equations are solved using a sixth-order accurate compact (Pade´) finitedifference scheme for evaluation of the spatial derivatives [17] in all of the three directions. The finite-difference scheme allows flexibility in the specification of boundary conditions for minimal loss of accuracy relative to spectral methods. It is of sixth-order at inner points, of fourth-order at the next to boundary points, and of third-order at the boundary. In this Pade´ 3/4/6 scheme, the sixth-order accuracy at the inner points is achieved by a compact finite differencing. For a general variable /j at grid point j in the ydirection, the scheme can be written in the following form for the first and second derivatives: /0j1 þ 3/0j þ /0jþ1 ¼

ðET þ pÞw þ qz  usxz  vsyz  wszz   2 2 2 In Eqs. (2)–(5), ET ¼ q e þ u þv2 þw is the total energy

qx ¼ 

qT . cM 2

261

/00j1 þ

11 00 / þ /00jþ1 2 j

7 /jþ1  /j1 1 /jþ2  /j2 þ ; 3 12 Dg Dg /jþ1  2/j þ /j1 ¼6 Dg2 3 /jþ2  2/j þ /j2 þ . 8 Dg2

ð7Þ

ð8Þ

In Eqs. (7) and (8), Dg is the mapped grid distance in the ydirection which is uniform in space (grid mapping occurs when non-uniform grid is used), / can be any variable in the solution vector U defined in Eq. (2). More details on the Pade´ 3/4/6 scheme can be found in Reference [17]. Solutions for the discretized equations are obtained by solving the tridiagonal system of equations. The timedependent governing equations are integrated forward in time using a fully explicit low-storage third-order Runge– Kutta scheme [35]. Boundary conditions for the 3D spatial DNS of impinging jets represent a challenging problem. The 3D computational domain is bounded by the inflow and wall boundary in the streamwise direction, and open boundaries with the ambient field in the jet radial (cross-streamwise) direction. Physical conditions at the jet nozzle exit and the wall must be appropriately represented in the numerical simulation. In the meantime, boundary conditions in the jet crossstreamwise direction should allow jet mixing with the ambient fluid and entrainment. In this study, the Navier–Stokes characteristic boundary condition (NSCBC) by Poinsot and Lele [25] has been utilized for the inflow and wall boundary conditions, while non-reflecting characteristic boundary condition [29] has been used to specify open boundaries with the ambient field. The non-reflecting boundary condition allows the wall boundary layer flow going out of the computational domain and entrainment of the ambient fluid without causing spurious wave reflections.

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The specification of boundary conditions at the wall and domain inlet has been performed using the general formulation of boundary conditions for DNS, the NSCBC [25], which is based on the analysis of characteristics. The NSCBC associated with the high-order non-dissipative numerical algorithms uses the correct number of boundary conditions required for well posedness of Navier–Stokes equations that can avoid numerical instabilities and spurious wave reflections at the computational boundaries. In NSCBC, the local one-dimensional inviscid (LODI) relations are used to provide compatible relations between the physical boundary conditions and the amplitudes of characteristic waves crossing the boundary. The wall is assumed to be at constant ambient temperature, which is impermeable and satisfying the no-slip condition. The amplitudes of characteristic waves at the wall are estimated by the LODI system for walls [25], where the near-wall density has been treated as a ‘‘soft’’ variable that is allowed to change with the characteristic wave variations at this boundary. For the boundary condition at the jet nozzle exit (domain inlet), the NSCBC has also been used to specify the inflow boundary with density treated as a ‘‘soft’’ variable that varies slightly in the simulations according to the characteristic waves at the boundary. At the domain inlet (jet nozzle exit), the streamwise mean velocity has been specified as a hyperbolic tangent  ¼ f1 þ tanh½10ð0:5  rÞg=½1 þ tanhð5Þ profile, given by w qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 with r ¼ ðx  0:5Lx Þ þ ðy  0:5Ly Þ , where r stands for the radial direction of the round jet, originating from the center of the inlet domain (0 6 x 6 Lx, 0 6 y 6 Ly). The coefficients used in the above equation describing the velocity profile define the jet initial momentum thickness [22]. For the normal impingement with an incident angle of 90, the cross-streamwise mean velocity components at the domain inlet z = 0 are given by  u ¼ 0 and v ¼ 0. For the impinging jet with an external perturbation, an unsteady disturbance in a sinusoidal form is artificially added to the mean velocity profile at the domain inlet (jet nozzle exit). The velocity components at the domain inlet z = 0 for the impinging jets with an external perturbation are given by u¼ u þ A sinð2pf0 tÞ;  þ A sinð2pf0 tÞ. w¼w

v ¼ v þ A sinð2pf0 tÞ; ð9Þ

In Eq. (9), the amplitude of disturbance A is varied between  . The non-dimen0% and 4% of the maximum value of w sional frequency (Strouhal number) of the unsteady disturbance is f0 = 0.30, which has been chosen to be the unstable mode leading to the jet preferred mode of instability [12]. The inlet temperature profile is specified by the Crocco– Busemann relation [34]. For the heated impinging round jet, the temperature profile at the domain inlet has been kept unchanged during the simulations. Initially the pressure field is assumed to be uniform. In the simulations performed, the flow field is initialized using velocity and

temperature fields that vary linearly between their values at the domain inlet and those at the wall boundary. The initial conditions specified did not affect the numerical solutions appreciably after the initial stage of the flow development. 3. Numerical results and discussion Several computational cases have been performed with different upstream velocity conditions to examine the effects of an external perturbation on the dynamics of impinging jets. The code used in this study has been previously tested and validated for DNS of non-reacting and reacting open-boundary and sidewall-bounded jets and plumes [13–16]. In the following sections, results will be shown for four cases that constitute a comparative study: case A—a baseline case without an external perturbation; case B—a perturbed case with the perturbation amplitude A = 1%; case C—a perturbed case with A = 2%; and case D—a perturbed case with A = 4%, respectively. The boundary and initial conditions used have been defined in the previous section. In the simulations, the considered jet Mach number is M = 0.3, which is based on the centerline streamwise mean velocity at the domain inlet (jet nozzle exit) and other reference quantities specified in the previous section. The Reynolds and Prandtl numbers used in the simulations are Re = 1000 and Pr = 1. The Reynolds number under investigation is the limit at which a circular impinging jet becomes a transitional one [32], and the unit Prandtl number was used for simplicity. The ratio of specific heats used is c = 1.4. For the hot jet considered, there is a temperature ratio of T0/Ta = 2 at the inlet, where subscripts 0 and a represent the center of the domain inlet (jet nozzle exit) and the ambient environment respectively. The wall temperature is Tw = 1. The dynamic viscosity is chosen to be temperature-dependent according to l = la(T/ Ta)0.76 with the reference viscosity taken to be the ambient value. The dimensions of the computational box used are Lx = Ly = 12 and Lz = 6. The grid system used is of 180 · 180 · 120 nodes with a uniform distribution in each direction. In this study, a grid independence test was performed and further refinement of the grid to 240 · 240 · 120 did not lead to appreciable changes in the flow field solution. It is worth noting that the grid resolution used in this study is higher than that of the DNS of impinging round jet in the literature [31], where results for a higher Reynolds number impinging jet were reported using a lower-order numerical scheme. In the simulations performed in this study, the time step is limited by the Courant–Friedrichs–Lewy (CFL) condition for stability. The CFL number used is 2.0, which has been tested to give time-step independent results. Parallel computations have been performed on a 16-processor Beowulf Cluster computer under the MPI environment. The results presented next are considered to be grid and time-step independent, which will be discussed in terms of the instantaneous flow

X. Jiang et al. / Computers & Fluids 36 (2007) 259–272

and temperature fields, flow oscillatory behaviour, and time-averaged flow and heat transfer properties of the impinging jet. 3.1. Instantaneous flow and temperature fields Fig. 1 shows the instantaneous isosurfaces of vorticity magnitude ðx2x þ x2y þ x2z Þ1=2 at t = 80 of the four cases, where the three components of vorticity are xx = ow/oy  ov/oz, xy = ou/oz  ow/ox, xz = ov/ox  ou/oy, respectively. From the 3D plots, it is evident that the impinging jet deflects from the wall and then convects along the surface of the wall. The jet impingement forms a stagnation point, where the flow has zero velocity. The vorticity maxima occur at locations near the stagnation region. The maximum vorticity corresponds to locations with the strongest velocity deflection. In Fig. 1, the most important feature is that vortical structures are observed for the perturbed cases. Vortical structures are important features of impinging jets, as reported in the literature, e.g. [2,4,5,31]. The primary vortices in the primary jet stream caused by the Kelvin–Helmholtz type shear layer instability are evident for case D with 4% perturbation, which are also observable for case C with 2% perturbation.

263

For case B with 1% perturbation, the vortices are less obvious than cases C and D. There is also a lack of symmetry for the perturbed cases in Fig. 1, which is due to the asymmetric nature of the external perturbation. For jets, it is well known that there exists a preferred frequency at which an axisymmetric disturbance receives maximum amplification in the jet column and the jet develops large-scale vortical structures [12]. Fig. 2 shows the instantaneous temperature contours in the x = 6 plane at t = 80 of the four cases, corresponding to the 3D plots shown in Fig. 1. The vortical structures in the jet shear layers are observable for the three perturbed cases. From Fig. 2, it is obvious that the vortical level of the impinging jet is strongly affected by amplitude of the external perturbation, and a larger perturbation leads to a more vortical flow field. The Reynolds number under investigation is the limit at which an impinging jet becomes a transitional one [32]. The vortical characteristics and the significant velocity and temperature fluctuations observed in the flow fields (shown in Figs. 10, 11 and 13) indicate the transitional behaviour of the impinging jet when an external perturbation is present at the jet upstream locations. Flow vortical structures greatly affect the instantaneous flow and temperature fields. Figs. 3 and 4 show the

Fig. 1. Instantaneous isosurfaces of vorticity magnitude at t = 80 of the four cases (15 isosurfaces between the minimum and maximum values).

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Fig. 4. Instantaneous streamwise velocity profiles at t = 80 along the jet centerline (x = 6, y = 6) of the four cases.

Fig. 2. Instantaneous temperature contours in the x = 6 plane at t = 80 of the four cases (15 contours between the minimum and maximum values).

Fig. 3. Instantaneous temperature profiles at t = 80 along the jet centerline (x = 6, y = 6) of the four cases.

instantaneous temperature and streamwise velocity (zvelocity component w) profiles at t = 80 along the jet centerline of the four cases, respectively. It can be observed that the external perturbations lead to spatial oscillations in both the temperature and velocity profiles. For the perturbed cases, the centerline temperature and velocity do not change monotonically from the jet nozzle conditions to those at the wall. Larger external perturbations lead to larger amplitudes of the spatial oscillations. For case D with 4% external perturbation, the spatial oscillations at the jet centerline can be as high as 20% for the streamwise velocity and 5% for the temperature. From Fig. 3, it can be seen that the temperature gradients of the impinging jet near the wall are very large, where the temperature drops very sharply from its value in the primary jet stream to the value of the wall. The velocity profile shown in Fig. 4, meanwhile, does not drop as sharply as the temperature profile in the near wall region. This is because the thermal boundary layer near the wall starts its formation in the stagnation region, while the jet velocity deflects nearby and spreads in the cross-streamwise direction near the wall surface. For the impinging jet, the primary jet stream is deflected near the stagnation region, which is the starting location of the wall boundary layer. Vortical structures enhance mixing of the hot jet with the ambient fluid. Contours of the instantaneous crossstreamwise velocity (y-velocity component v) in the x = 6 plane at t = 80 of the four cases are shown in Fig. 5, which indicate velocity distributions of the wall boundary layer of the impinging jet. The formation of the wall boundary layer is due to the deflection of the primary jet stream near the stagnation region. The wall boundary layer is referred to as ‘‘wall jet’’ in the literature [1,2,5,18,23]. From Fig. 5, it is observed that the wall jet is constricted within a thin layer near the wall. Apart from the head vortex at the end of the wall jet, the velocity of the wall jet is unidirectional on either side of the stagnation region. There is no vortex existing in the wall boundary layer. The contour plots in Fig. 5 also indicate the vortical structures in the primary jet stream of the perturbed cases, where positive and negative cross-streamwise velocities co-exist. These vortical structures can enhance the entrainment of the primary jet stream and its mixing with the ambient fluid.

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Fig. 5. Instantaneous cross-streamwise velocity contours in the x = 6 plane at t = 80 of the four cases.

265

From Fig. 5, it also can be seen that the maximum wall jet velocities are about half of that of the primary jet stream. For the Reynolds number of the impinging jet, this velocity magnitude indicates that the wall jet is predominately laminar, unlike the transitional primary jet stream of the perturbed cases. For the study of impinging jets, wall stresses are of importance, particularly to near-wall modelling [6,7,24]. Fig. 6 shows the instantaneous wall shear stress syz and normal stress szz in the x = 6 plane at t = 80 of the four cases, while Fig. 7 shows the instantaneous wall shear stress syz contours at t = 80 of cases A and D. It can be seen that the largest normal stresses (negative value) occur in the stagnation point, where the shear stresses are of a zero value. The maximum wall shear stress occurs at locations close to the stagnation region and it has opposite signs on different sides of the impinging point due to the change of flow direction. The maximum shear stresses locate at the velocity deflection points, within one diameter to the impinging point. From Fig. 6, it also noticed that the shear stress is much larger than the normal stress. The maximum shear stress is approximately 30 times of that of the normal stress. In Fig. 6, the differences between the four cases are not significant. This is due to the fact that the near wall flow structures of the four computational cases do not differ each other significantly, as shown in Fig. 5 for the velocity contours. The external perturbation mainly affects the flow structures in the primary jet stream, while it does not have a significant impact on the wall boundary layer after the impingement. Finally, Fig. 6 shows that as the strength of the external perturbation increases, the symmetry around the stagnation point is lost gradually. This is especially clear in Fig. 7 in the shear stress distributions of cases A and D. The trend is an important indicator of the tendency towards transition to turbulence, which is usually absent from two-dimensional simulations. Fig. 8 shows comparison of the instantaneous Nusselt number at the wall in the x = 6 plane at t = 80 of the four cases. The Nusselt number is a dimensionless number which measures the enhancement of heat transfer from a

Fig. 6. Instantaneous wall shear and normal stresses (syz and szz) in the x = 6 plane at t = 80 of the four cases.

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Fig. 7. Instantaneous wall shear stress syz contours at t = 80 of cases A and D.

Fig. 8. Instantaneous Nusselt number at the wall in the x = 6 plane at t = 80 of the four cases.

surface which occurs in a ‘‘real’’ situation, compared to the heat transfer that would be measured if only conduction could occur. It is defined as Nu = hD/k, where D is the jet nozzle diameter and k is the thermal conductivity of the fluid and h is the heat transfer coefficient defined as h = k(dT/dz)/(T0  Tw). Typically Nusselt number is used to measure the enhancement of heat transfer when convection takes place. The instantaneous Nusselt number in Fig. 8 shows a ‘‘bell shape’’ distribution, which was reported extensively in the literature, e.g. [2,4,5,19,32]. A close examination of Fig. 8 reveals that the instantaneous Nusselt number fluctuates at regions near y = 4 and y = 8 for the perturbed cases, where the Nusselt number distribution shows small troughs and peaks. This is mainly because of the secondary vortices in the perturbed cases which arise due to the interaction between the primary vortices and the wall jets. The unsteady vortex separation from the wall in the perturbed cases leads to variations in the Nusselt number distribution due to the changes in the thermal layer thickness accompanying the unsteady flow structures. 3.2. Oscillatory behaviour of the impinging jet Oscillatory impinging jets may be used to achieve the desired heat and/or mass transfer rates in engineering

Fig. 9. Oscillatory behaviour of the impinging jet: instantaneous Nusselt number at the wall in the x = 6 plane at t = 80, t = 81.1, t = 82.2, and t = 83.3 of case D.

applications [27,28]. Oscillatory behaviour of the impinging jet has been observed for the perturbed impinging jet, such as the spatial oscillations shown in the jet centerline temperature and velocity profiles (Figs. 3 and 4). The oscillatory behaviour is associated with the vortical structures in the primary jet stream of the perturbed cases, which can lead to spatial and temporal variations of the flow properties. Fig. 9 shows the instantaneous Nusselt number at the wall in the x = 6 plane at t = 80, t = 81.1, t = 82.2, and t = 83.3 of case D respectively, which shows the temporal oscillatory behaviour of the impinging jet. The variations of the Nusselt number with time are mainly caused by the vortices in the primary jet stream originating from the jet nozzle exit, which impinge on the wall surface. In Fig. 9, the profiles at t = 80 and t = 83.3 are almost overlapping, due to the fact that this time interval approximately corresponds to one period of the external perturbation with f0 = 0.30. The influence of the secondary vortices, which arise due to the interaction between the primary vortices and the wall jet on the Nusselt number distribution, can also be seen in Fig. 9, where the instantaneous Nusselt number fluctuates at locations around y = 4 and y = 8.

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267

Fig. 10. Oscillatory behaviour of the impinging jet: time traces of the vorticity maxima of the four cases.

Vortical structures in the primary jet stream and their interactions with the wall are the reasons behind the flow oscillations. The flow vortical level can be represented by vorticity, which has three components for the 3D Cartesian coordinate system used. Fig. 10 shows time traces of the vorticity maxima of the four cases for the period between t = 20 and t = 80 after the initial stage of flow development. The impinging jet shows a periodically quasi-steady state during this period. For the unperturbed case A, the cross-streamwise vorticity is due to the velocity profile specified at the jet nozzle exit which does not change appreciably after the initial developing stage. For the perturbed cases, however, the maximum vorticity changes with time. It is also noticed that magnitude of the variation of the maximum vorticity changes with time, due to the interaction between the vortices in the primary jet stream and the wall. From Fig. 10, it is observed that the flow is dominated by the two cross-streamwise vorticity components xx and xy, while the streamwise vorticity component xz is insignificant. This is mainly because the flow tends to be re-laminarized in the wall boundary layer. An important phenomenon associated with the jet impingement is the re-laminarization effect of the wall. Upon impingement, the jet streamwise velocity decays to a zero value. In the meantime, the jet spreads over the wall

surface in the circumferential direction that forms the wall boundary layer. Since the wall boundary layer has a much larger cross-sectional area than that of the jet primary stream, the jet velocity decays significantly after the impingement. This consequently leads to the re-laminarization of the flow. Due to this re-laminarization effect of the wall, the impinging jet does not involve strong rotational motion in the streamwise direction as indicated by the insignificant xz in Fig. 10. Also due to this re-laminarization effect, the external perturbation does not have a significant effect on flow structures of the wall boundary layer as discussed previously. The external perturbation is responsible for the formation of vortical structures in the primary jet stream and for the oscillatory behaviour of the impinging jet, but it has insignificant effect on the wall boundary layer due to the re-laminarization effect of the wall. Fig. 11 shows time traces of the velocities at (x = 6, y = 6, z = 2) and (x = 6, y = 6, z = 4) along the jet centerline of the four cases, from which the temporal oscillation of the impinging jet is evident. For the unperturbed case, the velocity is a constant at these two locations. For the three perturbed cases, the external perturbation has apparently led to significant temporal variations in the instantaneous streamwise velocity. The instantaneous velocity changes with time in a sinusoidal mode, similar to the

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Fig. 11. Oscillatory behaviour of the impinging jet: time traces of the velocities at (x = 6, y = 6, z = 2) and (x = 6, y = 6, z = 4) of the four cases.

external perturbations applied at the domain inlet. For case D with 4% external perturbation, the variation of the instantaneous velocities at these two points along the jet centerline can be as high as 50% of the mean velocity. Apparently the external perturbation has been amplified in the jet column. The perturbation applied at the jet nozzle exit propagates along the primary jet stream and reflects at the wall, experiencing amplification due to the wave interaction. The continuous supply of perturbation leads to multiple reflections that interact with each other, which subsequently lead to large oscillations in the primary jet stream. A comparison between Fig. 11(a) and (b) also reveals that the velocity variation at z = 4 is higher than z = 2, due to the stronger wave interaction and amplification. To further clarify the amplitude of velocity oscillation in the primary jet stream, Fig. 12 shows the Fourier spectra of the velocities at (x = 6, y = 6, z = 2) and (x = 6, y = 6, z = 4) of the three perturbed cases, corresponding to the velocity histories shown in Fig. 11. It is observed that larger external perturbations at the jet nozzle exit leads to larger amplitudes of the velocity oscillation in the primary jet stream. The amplitude of the velocity oscillation is more than doubled when the external perturbation amplitude is doubled. From Fig. 12, it is clear that the oscillation of

the perturbed impinging jet is dominated by the frequency of the external perturbation that leads to the Kelvin–Helmholtz type shear layer instability. The flow does not contain appreciably higher frequencies. This is mainly due to the fact that the flow has insignificant streamwise vorticity xz as that shown in Fig. 10, and therefore there is no significant vortex breakdown and small scales in the flow field. Due to the coupling between momentum and energy transfer, the oscillatory behaviour occurs for both the velocity and temperature fields. Fig. 13 shows time traces of the temperatures at (x = 6, y = 6, z = 2) and (x = 6, y = 6, z = 4) along the jet centerline of case C with 2% external perturbation. It can be seen that temperatures at both locations vary significantly with time, indicating an appreciable temporal oscillation, similar to that observed for the velocity. In the meantime, it is noticed that the temperature at z = 4 is generally lower than that at z = 2, due to the mixing of the hot jet with the ambient fluid. It is also noticed that there is a phase difference between the temperature variations at the two locations. This is mainly because of the convection of vortical structures in the primary jet stream associated with the spatial and temporal evolution of the impinging jet. The vortical structures formed due to the Kelvin–Helmholtz shear layer instability are continuously convected downstream by the mean flow.

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Fig. 12. Oscillatory behaviour of the impinging jet: Fourier spectra of the velocities at (x = 6, y = 6, z = 2) and (x = 6, y = 6, z = 4) of cases B, C and D.

Fig. 13. Oscillatory behaviour of the impinging jet: time traces of the temperatures at (x = 6, y = 6, z = 2) and (x = 6, y = 6, z = 4) of case C.

Fig. 15. Time-averaged streamwise velocity profiles along the jet centerline (x = 6, y = 6) of the four cases.

3.3. Time-averaged flow and heat transfer characteristics

files along the jet centerline (x = 6, y = 6) of the four cases. They correspond to the instantaneous profiles shown in Figs. 3 and 4, respectively. It can be seen that the timeaveraged temperature and streamwise velocity profiles do not differ significantly among the four cases. This is mainly because the external perturbations applied at the jet nozzle exit are periodic. Although the instantaneous flow and temperature fields can differ significantly among the four cases, the differences in time-averaged quantities become less. However, it is still noticeable that vortical structures enhance the mixing between the hot jet and its surroundings that lead to faster decay of the temperature. From Figs. 14 and 15, it is observed that the temperature gradients of the impinging jet near the wall are very large, while the velocity does not drop as sharply as the temperature profile in the near wall region. Fig. 16 shows the time-averaged wall shear and normal stresses (syz and szz ) in the x = 6 plane of the four cases, corresponding to the instantaneous plots shown in Fig. 6. From Fig. 16, it can be seen that differences between the four cases are insignificant. This is because of the re-laminarization effect of the wall that leads to similar wall jet structures for the four cases under investigation, as discussed previously for the instantaneous quantities. Corresponding to the instantaneous distributions, the shear stresses are zero at the stagnation point, where the largest

In this study, time averaging of the simulation results has been performed to examine the time-averaged flow properties of the impinging jet. The time interval used for the averaging is between t1 = 60 and t2 = 80, after the flow has reached a developed periodic stage. This time interval used for the averaging contains six periods of the external perturbation used for the three perturbed cases B, C and D. Fig. 14 shows the time-averaged temperature profiles along the jet centerline (x = 6, y = 6) of the four cases, while Fig. 15 shows the time-averaged streamwise velocity pro-

Fig. 14. Time-averaged temperature profiles along the jet centerline (x = 6, y = 6) of the four cases.

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Fig. 16. Time-averaged wall shear and normal stresses (syz and szz ) in the x = 6 plane of the four cases.

Fig. 17. Time-averaged Nusselt number at the wall in the x = 6 plane of the four cases.

normal stresses (negative value) occur. The largest wall shear stress occurs at the velocity deflection region within one diameter to the impinging point. The maximum shear stresses are much larger than the maximum normal stresses. It is observed that the external perturbation does not affect the time-averaged wall stresses appreciably. To further examine the wall heat transfer characteristics, Fig. 17 shows the time-averaged Nusselt number at the wall in the x = 6 plane of the four cases, corresponding to the instantaneous Nusselt number shown in Fig. 8. It is well known that the Nusselt number for wall heat transfer has a ‘‘bell shape’’ distribution, where the stagnation Nusselt number is predominately determined by the flow Reynolds number and the nozzle-plate distance plays a secondary role [2,19,32]. In Fig. 17, the time-averaged Nusselt number for all the cases shows a ‘‘bell shape’’ distribution. The stagnation Nusselt number for case A without perturbation is approximately 25. For the three perturbed cases, the stagnation Nusselt number is slightly lower and it also decreases slightly with increasing amplitude of the external perturbation. This is due to the fact that vortical structures induced by the perturbation in the primary jet stream enhance the mixing of the hot jet with the ambient fluid, which leads to lower temperature gradients near the wall. The asymmetric time-averaged Nusselt number of case D shown in Fig. 17 might be due to the insufficient averaging time for this case.

Results shown in Fig. 17 indicate that oscillations induced by the external perturbation slightly degrade heat transfer to the wall. Mixed results were reported in the literature [27,28], where it was understood that pulsation does not always improve the transfer rate: the pulses do not reach up to the wall. In the studies performed here, the external perturbation does not affect the wall boundary layer appreciably due to the re-laminarization effect of the wall, although it has significant effects on flow structures in the primary jet stream of the impinging flow. This is a case that the external perturbation does not reach up to the wall boundary layer; therefore the wall heat transfer is not enhanced by the oscillation. However, there are situations where periodic unsteadiness of the nozzle flow can enhance impinging jet heat and/or mass transfer [27,28], if the pulsations reach the wall boundary layer. Meanwhile, the vortical structures induced by the external perturbation can certainly enhance the fuel/air mixing in the primary jet stream of an impinging reacting flow that is encountered in the fuel injection process of a variety of industrial combustors. Most of the experimental data available in the literature are for high Reynolds number impinging jets [19,32]. In the direct numerical simulations performed in this study, the Reynolds number achieved was restricted by the computational resources available. Due to the relevance to practical applications, an impinging jet with moderate distance-todiameter ratio rather than a very shallow configuration or a large-distance impinging configuration was investigated in this study. The closest experimental conditions in the literature are for a case with Reynolds number 1000 and a shallow configuration with distance-to-diameter ratio 2, where the stagnation Nusselt number was obtained to be around 31 [2]. This experiment value is higher than the stagnation Nusselt number obtained in this study with larger nozzle-to-plate distance. However, it is well known that the Nusselt number increases with a decrease in the nozzle-to-plate distance [19,32]. Therefore the Nusselt number obtained in this study is expected to be reasonable.

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The profile of the Nusselt number distribution is also very similar to those experimental and numerical profiles reported in the literature [2,4,5,19,32]. 4. Concluding remarks Detailed characteristics of the heated impinging round jets have been numerically explored by solving the compressible Navier–Stokes equations using highly accurate numerical methods. Effects of an external perturbation on the flow and temperature fields of the impinging jets have been examined by a comparative DNS. The simulations performed focused on a relatively low Reynolds number Re = 1000 and a moderate nozzle-to-plate distance of 6 jet diameters. Both instantaneous and time-averaged simulation results have been presented for the four computational cases, including a case without an external perturbation, and three perturbed cases with velocity perturbation amplitudes of 1%, 2% and 4%, respectively. The simulation results have revealed that the unsteadiness associated with the external perturbation leads to vortical structures in the jet shear layers emerging from the nozzle. These vortical structures occur in the primary jet stream. Due to the interaction with the wall, these vortical structures lead to significant spatial and temporal oscillations of the impinging jet. The flow oscillation amplitude increases with increasing external perturbation amplitude. With 4% velocity perturbation, the spatial oscillation of the jet centerline velocity can be as high as 20%, while the temporal oscillation can be as high as 50%. The amplitude of these oscillations decreases with decreasing amplitude of the external perturbation. Vortical structures that are induced by the external perturbation also enhance the mixing between the hot jet and the ambient fluid. The computational cases involving the external perturbations show transitional behaviour in the primary jet stream, while the unperturbed case does not have a transitional tendency. For the perturbed cases, the flow can be re-laminarized by the impingement. Details of the near-wall stresses and heat transfer characteristics have been obtained, which can be useful to near-wall model development. It has been observed that the wall shear stresses are much larger than the normal stresses, where the peak value of the former is about 30 times of the latter for the impinging configuration investigated. The Nusselt number distribution obtained from the direct simulation is in agreement with the available experiment for similar conditions in the literature. It has been found that the external perturbation does not play a significant role in the near-wall stresses and heat transfer characteristics. This is because the vortical structures in the primary jet stream do not change the near wall flow structures significantly due to the re-laminarization effect of the wall. For the Re = 1000 impinging jet investigated, the perturbed cases show transitional behaviour in the primary jet steam. However, the deflection of the primary jet stream at the wall leads to a lower velocity laminar wall jet,

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