DNS of compressible turbulent boundary layers at ...

AIAA 2017-3116

DNS of compressible turbulent boundary layers at varying subsonic Mach numbers Christoph Wenzel∗ , Bj¨orn Selent† , Markus J. Kloker‡ and Ulrich Rist§ Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, 70550 Stuttgart, Germany

Direct numerical simulations of compressible zero pressure gradient turbulent boundary layers are performed for varying subsonic Mach numbers from M∞ = 0.3 up to M∞ = 0.85. Whereas the former represents a nearly incompressible flow, the latter is comparable to transonic free flight. For the quantification of the compressibility effects, all simulations are compared to incompressible results available in literature. Whereas the skin friction coefficient cf shows differences up to 5% in the simulated range of Mach numbers, the shape factor H12 differs about 20%. For both, the local mean velocity profiles and the Reynolds fluctuations only small compressibility effects can be observed.

Nomenclature δ∗ δ99 κ µ ν < ρ τw θ cf cp , cv E e H12 L N p Pr Reτ Reθ Rex Reδ,99 T t

displacement thickness boundary layer thickness heat capacity ratio dynamic viscosity kinematic viscosity universal gas constant density wall shear stress momentum thickness skin friction coefficient specific heats total energy edge of the boundary layer thickness at δ99 shape factor reference length grid point index pressure Prandtl number skin friction Reynolds number momentum thickness Reynolds number Reynolds number of streamwise position boundary layer thickness Reynolds number temperature time

u, v, w streamwise, wall-normal and spanwise velocity component uτ skin friction velocity x, y, z streamwise, wall-normal and spanwise direction Q dimensionless solution vector M Mach number Subscripts 0 ∞ i w c end inc max

inlet of the simulation domain farfield running index for x, y and z wall properties compressible variables outlet of main region of simulation domain incompressible variables outlet of whole simulation domain

Superscripts 0

∗ + T

fluctuating properties nondimensionalized values wall units time and space averaged properties transposed vector

∗ Research

Assistant, Institute of Aerodynamics and Gas Dynamics, [email protected]. Assistant, Institute of Aerodynamics and Gas Dynamics, [email protected]. ‡ Associate Professor, Institute of Aerodynamics and Gas Dynamics, Senior AIAA member, [email protected]. § Associate Professor, Institute of Aerodynamics and Gas Dynamics, [email protected]. † Research

1 of 10 American Institute of Aeronautics and Astronautics

I.

Introduction

For many years now, the canonical case of zero pressure gradient turbulent boundary layer (ZPGTBL) forms the essential foundation in turbulence research. Whereas fundamental understanding of the physical behaviour of wall-bounded turbulent flows was gained already for more than hundred years by experimental measurements and analytical investigations, a second and more closer view into turbulence was provided by the very first Direct Numerical Simulation (DNS) of incompressible ZPGTBL by Spalart in 1988.1 Even though the computational power has grown enormously since Spalarts pioneering work, DNS of TBLs are still strongly limited by the computational resources. While the early DNS of turbulent boundary layers (TBLs) could only consider incompressible flows by neglecting the energy equation because of computational costs, nowadays also compressible simulations are available in literature. Due to the additional differential equation and the Mach number dependence of the viscous time-step limit, the computational effort for unsteady simulations of compressible flows greatly exceeds that of incompressible flows. As a result of a less restricting viscous time-step limit and a significant change in flow behaviour for high Mach-number flows, DNS results of compressible TBLs are mainly available for supersonic or hypersonic flows in literature. In many applications, however, turbulent flow is neither incompressible nor supersonic. Even for objects moving relatively slow compared to sonic speed, the surrounding flow can be locally in the upper subsonic Mach number range due to the flow acceleration resulting from changes in geometry. Although only small deviations can be expected for middle range subsonic Mach numbers compared to incompressible flow, these are expected to be non-negligible for high subsonic Mach numbers, e.g. M∞ = 0.85. Despite its enormous relevance for air traffic, no DNS results performed with the spatial approach are known to the authors, which cover TBLs at high subsonic Mach numbers. With the objective of creating a reference for fundamental investigations in terms of flow control like uniform blowing (UB) and uniform suction (US), this work provides DNS results of compressible ZPGTBLs for varying subsonic Mach numbers at M∞ = 0.3, 0.5, 0.7 and 0.85. Whereas the first represents a nearly incompressible flow, the latter is comparable to cruise speed of commercial aircraft.

II. II.A.

Numerical methodology

Numerical method

All computations are performed with the compressible high order in-house DNS code N S3D, which has been extensively validated for a broad variety of applications. Fundamentals of the code are described in Babucke et. al.,2 Linn & Kloker3, 4 and Keller & Kloker.5, 6 It solves the three-dimensional, unsteady, compressible Navier-Stokes equations together with the continuity and energy equation in conservative formulation. AsT suming a velocity vector u = [u, v, w] in streamwise, wall-normal and spanwise directions x, y and z, the T dimensionless solution vector is Q = [ρ, ρu, ρv, ρw, E] , where ρ and E are the density and the total energy. ∗ Velocities and length scales are normalized by the streamwise reference velocity U∞ and the reference length ∗ ∗ L , respectively. Thermodynamical quantities are normalised with the reference temperature T∞ and the ∗ ∗ ∗2 reference density ρ∞ . Pressure p is non-dimensionalised by ρ∞ U∞ . The specific heats cp and cv as well as the Prandtl number P r are assumed to be constant. Temperature dependence of the viscosity is modeled by Sutherland’s law. The equations are solved on a block-structured Cartesian grid spanning a rectangular integration domain. The chordwise and wall-normal directions are discretised using 8th -order explicit finite differences,2 whereas the z-derivatives for the spanwise direction are calculated by a Fourier-spectral ansatz. The classical fourth-order Runge-Kutta scheme is used for time integration. II.B.

Numerical setup and boundary conditions

A sketch of the simulation domain is given in Fig. 1. In analogy to simulations available in literature, the main region (red framed in Fig. 1) of the simulation domain is designed with a height of at least three boundary layer thicknesses δ99,end , measured at the end of the main region xend . Due to the Fourier-spectral ansatz in spanwise direction, the spanwise dimension is set to π δ99,end . The flow field is periodic in spanwise direction. At the solid wall, the flow field is treated as fully adiabatic with (dT /dy)w = 0 which suppresses any heat exchange between wall and fluid, whereas the pressure at the wall is calculated by (dp/dy)w = 0 from the


Figure 1. Simulation domain for present DNS. Yellow colored regions represent sponge zones, blue colored regions represent grid-stretched and filtered regions. The red bordered zone represents the main region of the simulation.

interior field. For the velocity components, a no-slip boundary condition is applied. At the inlet a digital filtering synthetic eddy method (SEM) is used to generate an unsteady, pseudoturbulent inflow conditions.7–9 The required distributions for the mean flow field and for the Reynolds fluctuations are used from a small auxiliary simulation using a recycling/rescaling method at M∞ = 0.510–12 for all cases . Even if the SEM boundary condition provides a pseudo-physical turbulent flow field already at the inlet of the domain, the flow needs about 10δ99,0 in streamwise direction to satisfy equilibrium turbulent flow statistics. In order to prevent the farfield flow from being distorted by this transition process, a sponge region13 is applied in the inlet region of the simulation domain (yellow colored in Fig. 1), which damps down the flow to an unperturbed free stream baseflow. At the top of the domain all thermodynamical quantities ρ, T and p are kept constant. The velocity components u and w are specified by d/dy = 0, whereas the wall-normal velocity component v is computed from the continuity equation under the assumption of dρ/dy = 0 such that dv/dy = −1/ρ (d (ρu) /dx + d (ρw) /dz). At the outflow, the time derivative is extrapolated with ∂Q/∂t|N = ∂Q/∂t|N −1 . In both wall-normal and streamwise direction, the numerical grid is stretched and filtered14, 15 in order to avoid reflections from the boundaries. It was found that an additional sponge in the outflow region of the domain enhances the quality of the simulation results notably by fixing the farfield flow also in the outflow region. For the purpose of mass conservation, the outflow sponge begins outside of the boundary layer. II.C.

Simulation parameters

The main region of the computational box has a dimension of T∞ [K] 288.15 42 δ99,end × 3 δ99,end × π δ99,end in streamwise, wall-normal and spanPr [−] 0.71 wise direction, respectively. The use of an identical setup for all Mach 3 ρ∞ kg/m 1.225 number cases should allow a maximum of comparability between the different cases. Therefore, the domain size is chosen to cover the same R [J/ (mol K)] 287 range of ∆Rex = ρ∞ U∞ (xend − x0 ) /µ∞ and almost the same range κ [−] 1.4 of Reδ,99 = ρ∞ U∞ δ99 /µ∞ for all Mach numbers on an identically Table 1. Thermodynamic properspaced numerical grid in terms of ∆Rex,y,z . This setting ensures the ties. same amount and distribution of grid points at comparable boundary layer thicknesses in terms of Reδ,99 and spatial positions in terms p of ∆Rex . As the result of the increased viscosity in the vicinity of the wall, the friction velocity uτ = τw /ρw and thus the skin friction Reynolds number Reτ = ρw uτ δ99 /µw is slightly decreasing for higher Mach numbers at comparable Reδ,99 . The grid resolution is based on the most restricting, nearly incompressible case at M∞ = 0.3. The corresponding grid resolution is 1100 × 150 × 256 grid points for the main region (red framed in Fig. 1) and 1200 × 180 × 256 grid points for the overall domain. Calculated in viscous wall units, this gives a grid spacing of ∆x+ = 21, ∆y1+ = 0.8 and ∆z + = 6.9 at Reθ = 670 for M∞ = 0.3 and of ∆x+ = 19.5, ∆y1+ = 0.75 and ∆z + = 6.4 at Reθ = 670 for M∞ = 0.85 in the three dimensions. The averaging time after the initial transient is about ∗ 990 time units (δend /U∞ ) corresponding to ∆t+ = ∆tu2τ /νw ≈ 3000 in viscous units for M∞ = 0.3. Due to the less restricting viscous time step limit for higher Mach numbers, the averaging time is three times higher for M∞ = 0.85 after simulating the same number of time steps. Even if this is not enough to make sure that all the statistics are sufficiently converged especially for the M∞ = 0.3 case, important trends can


be estimated reliably. The basic thermodynamic flow parameters are shown in Tab. 1 and set equal for all cases.

III.

Results

This work focuses on the quantification of Mach number effects for compressible, subsonic TBLs. Therefore, only substantial mean values of the simulated results are presented in this study without focusing on high-order statistics like energy spectra or turbulent energy budgets. In the following, the spatial evolution of averaged mean flow statistics will be investigated at first, whereas local mean flow quantities are analyzed in the section afterwards. III.A.

Spatial evolution of averaged mean flow statistics

All results presented in the following are averaged both in time t and spanwise direction z. The compressible ¨ u16 results are compared to incompressible DNS data available in literature, summarized by Schlatter & Orl¨ and presented in Fig. 2. It should be noted that only regions of fully turbulent flow are presented here, whereas the inflow region and the outflow region of the simulation domain are not considered. The spatial development of the skin friction coefficient cf , defined as τw ∂U (1) τ = µ with cf = 1 w w 2 ∂y w 2 ρ∞ U∞ is given in Fig. 2(a), whereas the friction-velocity Reynolds number Reτ is shown in Fig. 2(b). The subscript w and ∞ denote values at the wall and in the farfield, respectively. The shape factor H12 , which is defined as the ratio between the displacement thickness δ ∗ and the momentum thickness θ, is given in Fig. 2(c) with ∗

δ =

Zδ99 0

ρU 1− ρe U e

dy

and

θ=

Zδ99 0

ρU ρe U e

1−

U Ue

dy.

(2)

Fluid properties evaluated at the edge of the boundary layer δ99 are denoted by the subscript e. All quantities are plotted versus the momentum thickness Reynolds number Reθ , representing the spatial streamwise length. The plot for the cf -distribution is supplemented by an incompressible correlation (black solid line in −1/4 17 Fig. 2(a)) based on the 1/7-power law of the form cf = 0.024 Reθ together with its compressibility transformed counterparts for the considered Mach numbers (grey short-dashed lines in Fig. 2(a)). These are calculated with van Driest’s compressibility transformation18 (often labeled as van Driest II transformation in literature) which can be written as 1 µe cf,c = Reθ,c . cf,inc (3) Fc µw The subscript c and inc distinguish between compressible and incompressible variables. In case of adiabatic flow, Fc and A are defined as Fc =

T w /T e − 1 , sin−1 A

A= q

T w /T e − 1

. T w /T e T w /T e − 1

(4)

The plot for the Reτ -distribution is supplemented by an incompressible best-fit power-law relation pro¨ u16 with Reτ = 1.13 Re0.843 (black solid line in Fig. 2(b)), whereas the shape factor vided by Schlatter & Orl¨ θ distribution H12 is supplemented by the shape of integrated incompressible composite profiles19 (black solid line in Fig. 2(c)). The blue and red colored dots denote the streamwise position of Reτ = 359.

III.A.1.

Flow quality of incompressible simulation

In this section, only the results for the nearly incompressible case at M∞ = 0.3 are considered. These are represented by a red solid line in Fig. 2. It should be mentioned that compressibility effects are not negligible in this case, since the temperature ratio T e /T w differs by about 2% compared to a truly incompressible case. 4 of 10 American Institute of Aeronautics and Astronautics

5.5 cf,Correlation

Reτ = ρw uτ δ99 /µw

cf × 103

5.0

4.5

4.0

00 1000 9 0 80 0 0 7 0 60 0 50 0 40 0 30

3.5 0 20

Reτ,Best f it

3.0 0

50

0

60

0 70

0

0 0 90 100 Reθ = ρ∞ U∞ θ/µ∞ 80

0 50

0

0 20

(a) Skin friction coefficient cf over momentum thickness Reynolds number Reθ .

0 60

0 70

0 80

0 0 90 100 Reθ = ρ∞ U∞ θ/µ∞

20

00

(b) Skin friction velocity Reynolds number Reτ over momentum thickness Reynolds number Reθ .

1.9

H12 = δ ∗ /θ

1.8

1.7

1.6

1.5

1.4

M∞ = 0.3 (inc. f ormulation) H12, Correlation

1.3 0

50

0

60

0

70

0 80

0

00 10 Reθ = ρ∞ U∞ θ/µ∞ 90

00

20

(c) Shape factor H12 over momentum thickness Reynolds number Reθ .

Legend: Detailed references can be found in Ref.16

Figure 2. Developement of turbulent mean flow quantities for cZPGTBLs at M∞ = 0.3, 0.5, 0.7 and 0.85, compared to incompressible results summarized in Ref.16

Nevertheless, the present results for the development of the skin friction coefficient cf is in good agreement ¨ u16 ( in Fig. 2) and the given with the most reliable incompressible DNS data provided by Schlatter & Orl¨ correlation for Reynolds numbers between Reθ = 600 and 1400 (Fig. 2(a)). Comparing the curve for Reτ to the incompressible references, a nearly perfect fit can be determined with the best fit regression (black solid line in Fig. 2(b)), whereas differences can be observed between the M∞ = 0.3 case and the data provided ¨ u16 for lower Reynolds numbers. Since these differences disappear for higher Reynolds by Schlatter & Orl¨ numbers, they could be traced back to history effects caused by different inflow boundary conditions which are applied in both simulations. Due to its dependency on sensitive parameters like the boundary-layer thickness δ99 and the skin-friction velocity uτ on the ordinate as well as the momentum thickness θ on the abscissa, Fig. 2(b) provides an honest measure for the flow quality. The shape factor H12 , given in Fig. 2(c) shows a systematical difference to the incompressible results by about 2%. Since the incompressible correlation and the M∞ = 0.3 case collapse when the shape factor is calculated with its incompressible formulation by neglecting the density variations across the boundary layer (red dashed line in Fig. 2(c)), this difference can be traced back to compressibility effects. As there is good agreement between the M∞ = 0.3 case and the incompressible data base, also the simulations for higher Mach numbers are expected to be highly reliable.


III.A.2.

Mach number dependency of compressible TBLs

Comparing the results for the Mach numbers M∞ = 0.5, 0.7 and 0.85 to M∞ = 0.3 in Fig. 2, significant differences can be observed. For all parameters shown the compressible results seem to obey a parallel shift of the incompressible results. Whereas these differences are about 5% in the cf -distribution for the case at M∞ = 0.85, the Reτ -distribution is decreased by about 13%. The highest Mach number influence can be detected for the shape factor H12 , which is increased for about 20% at M∞ = 0.85, about 12% at M∞ = 0.7 and still 5% at M∞ = 0.5 compared to the M∞ = 0.3 case. When the cf -distributions are compared to the gray short-dashed lines in Fig. 2(a), representing the compressibility transformed cf -distributions of the incompressible correlation (see Eq. 3), a systematical shift can be observed. Taking into account that this shift is caused by a mismatch between the incompressible correlation and the M∞ = 0.3 case, the compressibility effect on the cf -distribution can be predicted almost exactly by van Driest’s flat-plate theory.18 Apart from the shape-factor distribution it should be noted that all compressible results for the whole subsonic range lie in between the scattering of incompressible DNS data (black symbols in Fig. 2). Consequently it is advisable to simulate an independent incompressible reference case with a comparable setup, when compressibility effects should be quantified in the subsonic Mach number range. III.B.

Local flow statistics

In this section, local mean flow statistics are investigated applying the van Driest transformation. Additionally, local fluctuating statistics for the velocity field as well as for the temperature and density field are compared to each other. For the velocity fluctuations also Morkovin’s hypothesis will be investigated. Both the van Driest transformation and Morkovin’s hypothesis are introduced briefly in the following. It is extensively verified by experiment and DNS20–22 that the logarithmic layer of the streamwise component U of a compressible TBL can be transformed into its incompressible counterpart using the van Driest transformation.23 This transformation takes the density variation over the boundary layer in the wall normal direction into account 1 1/2 log y + + C, dU V D = (ρ/ρw ) dU . (5) κ According to incompressible data, the constants can be defined as κ = 0.41 and C = 5.2. The van Driest transformation is closely coupled with Morkovin’s hypothesis24, 25 which is anticipated in its derivation. Morkovin postulated that ”the essential dynamics of these shear flows will follow the incompressible pattern“ for moderate Mach numbers.26 If it is assumed that (i) turbulent time and length scales are not affected by compressibility and (ii) a constant stress layer exists across the wall layer, the shear 2 stress and normal-stress distributions −ρu0 v 0 / ρw u2τ and −ρu02 i / ρw uτ can be assumed to be invariant of the Mach number if they are compared at the same skin-friction Reynolds number Reτ .25, 27 The densityscaling of the stress distributions is often referred to as Morkovin transformation. Without doubt both the van Driest transformation and Morkovin’s hypothesis can be denoted as the most important relations for compressible wall-bounded flows. +

UV D =

III.B.1.

Local mean flow statistics +

Figure 3 gives the mean-flow velocity profiles in streamwise direction U in wall units at Reτ = 359, which ¨ u’s incompressible corresponds to Reθ = 951 in the M∞ = 0.3 case and Reθ = 1006 in Schlatter & Orl¨ dataset.16 For the M∞ = 0.5, 0.7 and 0.85 cases Reτ = 359 corresponds to Reθ = 988, 1044 and 1092, + respectively (see the blue and red colored dots in Fig. 2(b)). Whereas the mean-flow profiles U are colored + in red, the van Driest transformed effective velocity profiles U V D are represented by blue lines. The black ¨ u.16 The dashed black circled solid line represents the incompressible reference provided by Schlatter & Orl¨ + + lines denote the viscous sublayer with U = y + and the logarithmic law of the wall with U = 1/κ log y + +C with κ = 0.41 and C = 5.2. We would like to emphasize that the U + -profiles compared are read at different Reδ,99 , different Reθ and different Rex when they are compared at same Reτ , which makes the assessment of the identified Mach number effects very difficult to interpret in terms of the spatial position.


25

U + = U /uτ

20

Ref erence M∞ = 0.3 M∞ = 0.5 M∞ = 0.7 M∞ = 0.85

M∞,V D = 0.3 15

M∞,V D = 0.5 M∞,V D = 0.7

23

M∞,V D = 0.85

22 10 21 20

5

19 0 100

101

300

400

102 y

+

500 600 700 103

= yuτ /ν w

Figure 3. Local mean flow velocity profile in wall units for Reτ = 359 at M∞ = 0.3, 0.5, 0.7 and 0.85. Red lines + + denote results for compressible simulations U , van Driest transformed velocity profiles U V D are denoted by blue lines. ¨ u.16 Incompressible reference data are taken from Schlatter & Orl¨

+

When the velocity distribution U for M∞ = 0.3 (red solid line in the zoomed area in Fig. 3) is ¨ u’s data, good agreement can be observed. Since the van Driest transformed compared to Schlatter & Orl¨ + + velocity profile U V D (blue solid line in the zoomed area in Fig. 3) almost completely collapses with that of U , compressibility effects seem to be negligible for the M∞ = 0.3 case in the nondimensionalized representation. In the logarithmic region a comparison between the M∞ = 0.5, 0.7 and 0.85 cases and the M∞ = 0.3 case + shows only very small shifts towards smaller U -values with increasing Mach numbers. As already has been shown for higher Mach numbers (e.g. M∞ = 2.0 in Pirozzoli et al.22 ), this shift is reversed by the van Driest transformation, which collapses all simulations into the nearly incompressible distribution also for subsonic cZPGTBLs. + In the wake region however, quantifiable differences can be observed both for the velocity U and the van + Driest transformed effective velocity U V D when they are compared to the incompressible reference. Since the + velocity profiles are compared at same Reτ the wall-normal position of the boundary-layer edge y + = δ99 is + the same for all Mach numbers. Therefore, the velocity distributions U are only affected by compressibility + effects on the ordinate in Fig. 3. As the result of the increased uτ -values, the U -values are reduced for increasing Mach numbers. The maximum difference between the M∞ = 0.85 case and the incompressible + case is about 1% at y + = δ99 . + The blue lines representing the van Driest transformed velocity profiles U V D in Fig. 3 show a rising trend + + with increasing Mach numbers at the boundary-layer edge δ99 . Although the U V D -value for M∞ = 0.85 only exceeds the incompressible value by about 0.5%, this trend is in accordance with nearly all compressible experimental or - mainly temporal - DNS data available in literature. The most careful DNS of spatially evolving cZPGTBL available in literature in terms of grid resolution, inflow boundary condition and flow validation provided by Pirozzoli et al.,22 however, refutes this trend. They have shown that the van Driest ¨ u’s incompressible reference transformed velocity profiles for M∞ = 2.0 perfectly match Schlatter & Orl¨ data.16 For a clarification of this contradiction, also supersonic simulations are under way on an extended numerical grid. III.B.2.

Local fluctuating flow statistics

The distribution of the Reynolds fluctuations and the density-scaled Reynolds fluctuations (Morkovin transformed) are given in Fig. 4(a) and Fig. 4(b), respectively. Both are represented in wall units. The different Mach number cases are distinguished by a red solid line for the nearly incompressible case at M∞ = 0.3 ¨ u16 and blue lines for higher Mach numbers. The incompressible reference case provided by Schlatter & Orl¨


3.0

u+

2.0

1.5

w

ρ/ρw u‘+ i =

1.5

u‘+ i =

2.5

q

u+

q

2.0

Ref erence M∞ = 0.3 M∞ = 0.5 M∞ = 0.7 M∞ = 0.85

p

u‘2 i /uτ

2.5

ρ/ρw

Ref erence M∞ = 0.3 M∞ = 0.5 M∞ = 0.7 M∞ = 0.85

u‘2 i /uτ

3.0

+

1.0 v+

v+ 0.5

p

0.5

w+ 1.0

0.0 100

101

102 y

+

0.0 100

103

101

102

103

y + = yuτ /ν w

= yuτ /ν w

(a) Reynolds fluctuations u+ i .

(b) Density scaled Reynolds fluctuations (ρ/ρw )1/2 u+ i .

Figure 4. Local distribution of Reynolds fluctuations at Reτ = 359 in wall units for M∞ = 0.3 in red and M∞ = 0.5, 0.7 ¨ u.16 and 0.85 in blue. Incompressible reference data are taken from Schlatter & Orl¨

10 0.3 0.5 0.7 0.85

ρ‘2 /ρw × 102

= = = =

ρ‘+ × 102 =

T ‘+ =

4

2

0 100

M∞ M∞ M∞ M∞

2.0

= = = =

0.3 0.5 0.7 0.85

1.5

q

6

p

T ‘2