Although the actuator geometry is highly three-dimensional, the outer flowfield is nominally two-dimensional because of the high aspect ratio of the rectangular slot. For the present study, this configuration is modeled as a two-dimensional problem. A multi-block structured grid available at the CFDVAL2004 website is used as a baseline grid. The periodic motion of the diaphragm is simulated by specifying a sinusoidal velocity at the diaphragm surface with a frequency of 450 Hz, corresponding to the experimental setup. The amplitude is chosen so that the maximum Mach number at the jet exit is approximately 0.1, to replicate the experimental conditions. At the solid walls zero slip, zero injection, adiabatic temperature and zero pressure gradient conditions are imposed. In the external region, symmetry conditions are imposed on the side (vertical) boundaries and far-field conditions are imposed on the top boundary. A nominal free-stream Mach number of 0.001 is imposed in the free stream to simulate incompressible flow conditions in the TLNS3D code, which solves compressible flow equations. The code was run in unsteady (URANS) mode until the periodicity was established. The time-mean quantities were obtained by running the code for at least another 15 periods and averaging the flow quantities over these periods. The phase-locked average of flow quantities were assumed to be coincident with their values during the last full time period

Turkel, Eli

Vatsa, Veer N.

NASA Technical Reports Server

CASE 1: SIMULATION OF A SYNTHETIC JET IN QUIESCENT AIR
USING TLNS3D FLOW CODE
Veer N. Vatsa  and Eli Turkel
 Computational Modeling & Simulation Branch, NASA Langley Research Center, Hampton, VA
Tel Aviv University, Israel & NIA, Hampton, VA
Governing Equations and Solution Methodology
In the present work, a generalized form of the thin-layer Navier-Stokes equations is used to model the
turbulent, viscous flow encountered in a synthetic jet. The equation set is obtained from the complete Navier-
Stokes equations by retaining the viscous diffusion terms normal to the solid surfaces in every coordinate
direction. For a body-fitted coordinate system      fixed in time, these equations can be written in the
conservative form as:
J
 U
t

 F  F
v

 

 G G
v



 H H
v


  (1)
where U represents the conserved variable vector. The vectors F, G, H, and F
v
, G
v
, H
v
represent the
convective and diffusive fluxes in the three transformed coordinate directions, respectively. In Eqn. (1), J
represents the cell-volume or the Jacobian of coordinate transformation. A multigrid based, general purpose
multi-block structured grid approach is used for the solution of the governing equations. In particular,
the TLNS3D flow code is used in this study to solve Eqn. (1). Several references exist describing the
discretization and algorithmic details of TLNS3D code. We include only a brief summary of the general
features, and refer to the work of Vatsa and co-workers [1, 2] for further details regarding the TLNS3D code.
Numerical Discretization
The spatial terms in Eqn. (1) are discretized using a standard cell-centered finite-volume scheme. The con-
vection terms are discretized using second-order central differences with scalar/matrix artificial dissipation
(second- and fourth- difference dissipation) added to suppress the odd-even decoupling and oscillations in
the vicinity of shock waves and stagnation points [3, 4]. The viscous terms are discretized with second-order
accurate central difference formulas [1]. For the present computations, the Spalart-Allmaras (SA) model [5]
and the Menter’s SST model [6] are used for simulating turbulent flows.
For temporal discretization, the convective and diffusive terms are grouped together, and Eqn. (1) can
be rewritten as:
dU
dt
  C U D
p
 U D
a
 U (2)
where C U, D
p
 U, and D
a
 U are the convection, physical diffusion, and artificial diffusion terms,
respectively. The cell-volume or the Jacobian of coordinate transformation being included in these terms.
The time-derivative term can be approximated to any desired order of accuracy by the Taylor series
dU
dt


t
a

U
n 
 a
 
U
n
 a

U
n  
 a

U
n 
  (3)
1.4.1
https://ntrs.nasa.gov/search.jsp?R=20070031068 2019-08-29T18:38:50+00:00Z
Eqn. (3) represents a generalized backward difference scheme (BDF) in time, where the order of accuracy is
determined by the choice of coefficients a

, a
 
, a

... etc. For example, if a

 	, a
 
  
 and a

 	,
it results in a second-order accurate scheme (BDF2) in time, which is the primary scheme chosen for this
work due to its robustness and stability properties [7].
In Eqn. (3), the superscript n represents the time level at which the solution is known, and n+1 refers
to the next time level to which the solution will be advanced. Similarly, n-1 refers to the solution at one
time level before the current solution. Regrouping the time-dependent terms and the original steady-state
operator leads to the equation:
a

t
U
n 

E U
n n   

t
 S U
n
 (4)
where E Un n    depends only at solution vector at time levels n and prior, and S represents the steady
state operator or the right hand side of Eqn.(2). By adding a pseudo-time term, we can rewrite the above
equation as:
U


a

t
U
n 

E U
n n   

t
 S U
n
 (5)
Solution Algorithm
The algorithm used here for solving unsteady flows relies heavily on the steady-state algorithm available
in the TLNS3D code [1]. The basic algorithm consists of a five-stage Runge-Kutta time-stepping scheme
for advancing the solution in pseudo-time, until the solution converges to a steady-state. Efficiency of this
algorithm is enhanced through the use of local time-stepping, residual smoothing and multigrid techniques
developed for solving steady-state equations. In order to solve the time-dependent Navier-Stokes equations
(Eqn. 5), we added another iteration loop in physical time outside the pseudo-time iteration loop in TLNS3D.
For fixed values of S Un, E Un n   , we iterate on Un  using the standard multigrid procedure of
TLNS3D developed for steady-state, until desired level of convergence is achieved. This strategy, originally
proposed by Jameson [8] for Euler equations and adapted for the TLNS3D viscous flow solver by Melson
et. al [7] is popularly known as the dual time-stepping scheme for solving unsteady flows. The process is
repeated until the desired number of time-steps have been completed. The details of the TLNS3D flow code
for solving unsteady flows are available in Refs. [7] and [9].
Boundary Conditions
Except for the moving diaphragm, standard boundary conditions, such as the no-slip, no injection, fixed
wall temperature or adiabatic wall, far-field and extrapolation conditions are used as appropriate for the
various boundaries. The most accurate procedure to simulate the moving diaphragm would require moving
grid capability. For simplicity, we chose to simulate this type of boundary condition by a periodic tran-
spiration velocity condition. The frequency of the transpiration velocity at the diaphragm surface in the
numerical simulation corresponded to the frequency of the oscillating diaphragm, and the peak velocity at
the diaphragm surface was obtained from numerical iteration so as to match the experimentally measured
peak velocity of the synthetic jet emanating from the slot exit. The pressure at the moving diaphragm is also
required for closure. However, in the absence of unsteady pressure data from the experiment, we imposed
a zero pressure gradient at the diaphragm boundary. We also tested the pressure gradient boundary condi-
tion obtained from one-dimensional normal momentum equation [10], which had very little impact on the
solutions. Due to the simplicity and robustness, we selected the zero pressure gradient boundary condition
at the diaphragm surface.
1.4.2
Implementation and Case Specific Details
Although the actuator geometry is highly three-dimensional, the outer flow field is nominally two-dimensional
because of the high aspect ratio of the rectangular slot. For the present study, this configuration is modeled
as a two-dimensional problem. A multi-block structured grid available at the CFDVAL2004 website is used
as a baseline grid.
The periodic motion of the diaphragm is simulated by specifying a sinusoidal velocity at the diaphragm
surface with a frequency of 450 Hz., corresponding to the experimental setup. The amplitude is chosen so
that the maximum Mach number at the jet exit is approximately , to replicate the experimental conditions.
At the solid walls zero slip, zero injection, adiabatic temperature and zero pressure gradient conditions are
imposed. In the external region, symmetry conditions are imposed on the side (vertical) boundaries and
far-field conditions are imposed on the top boundary. A nominal free-stream Mach number of 0.001 is
imposed in the free stream to simulate incompressible flow conditions in the TLNS3D code, which solves
compressible flow equations.
The code was run in unsteady (URANS) mode until the periodicity was established. The time-mean
quantities were obtained by running the code for at least another 15 periods and averaging the flow quantities
over these periods. The phase-locked average of flow quantities were assumed to be coincident with their
values during the last full time period.
Results/Discussion
Only a brief set of results is included here, detailed results will be available in the workshop proceedings.
Majority of the solutions were obtained with SA turbulence model on the baseline grid with a time-step of
t  T
period

, where T
period
represents the physical time for a complete period. In addition, results
were generated for the following conditions: (a) coarser grid (cg) obtained by eliminating every other point
from the baseline grid; (b) finer grid (fg) generated by increasing resolution in the normal direction by 	;
(c) lower time-step (lower dt), where t  T
period
; and (d) Menter’s SST turbulence model. The
results for the vertical velocity near the slot exit center line (x=0., y=0.1 mm) from these computations are
compared in Fig. 1, which shows very little sensitivity to turbulence model change, and spatial and temporal
resolutions.
Next we show the results for time-averaged velocity along the jet center line in Fig. 2. In the proximity
of the slot exit (y  
 mm ), all 5 sets of results are indistinguishable, and are in good agreement with the
hotwire data. However, further away from the slot, the coarse grid (cg) results show significant deviation
from the baseline results. The effect of turbulence model type (SA or SST) are also more pronounced in the
outer region. Similar trends are observed in Fig. 3 for the jet width based on time-averaged solutions.
Based on these results and more detailed comparisons (not shown here) of the computational results
for time-averaged and phase-locked average velocities, the effect of lowering the time-step and refining the
normal mesh were found to be very small. Noticeable differences were observed for the coarse grid (cg)
results in the outer regions. The difference between the SA and SST turbulence model results become more
pronounced in regions further away from the slot exit.
References
[1] Vatsa, V.N. and Wedan, B.W.: “Development of a multigrid code for 3-d Navier-Stokes equations and
its application to a grid-refinement study”. Computers and Fluids. 18:391-403. 1990.
[2] Vatsa, V.N.; Sanetrik, M.D.; and Parlette, E.B.: “Development of a Flexible and Efficient Multigrid-
Based Multiblock Flow Solver”. AIAA paper 93-0677, Jan. 1993.
[3] Turkel, E. and Vatsa, V.N.: “Effect of artificial viscosity on three-dimensional flow solutions”. AIAA
Journal, vol. 32, no. 1, Jan. 1994, pp. 39-45.
1.4.3
[4] Jameson, A.; Schmidt, W.; and Turkel, E.: “Numerical Solutions of the Euler Equations by Finite
Volume Methods Using Runge-Kutta Time-Stepping Schemes”. AIAA Paper 1981-1259, June 1981.
[5] Spalart, P.A. and Allmaras, S.R.:“A one-equation turbulence model for aerodynamic flows”. AIAA
Paper 92-439, Reno, NV, Jan. 1992.
[6] Menter, F.R.: “Zonal Two Equation k-	 Turbulence Models for Aerodynamic Flows”. AIAA paper
93-2906, Orlando, Fl, 1993.
[7] Melson,N.D.; Sanetrik, M.D.; and Atkins, H.L.:“Time-accurate calculations with multigrid acceler-
ation”. Appeared in “Proceedings of the Sixth Copper Mountain conference on multigrid methods”,
1993, edited by: Melson, N.D.; Manteuffel, T.A.; and S.F. McCormick, S.F.
[8] Jameson, A.: “Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows
past Airfoils and Wings”. AIAA Paper 91-1596, 1991.
[9] Bijl, H.; Carpenter, M.H.; and Vatsa, V.N.: “Time Integration Schemes for the Unsteady Navier-Stokes
Equations”. AIAA Paper 2001-2612.
[10] Rizzetta, D.P.; Visbal. M.R.; and Stanek, M.J.: “Numerical Investigation of Synthetic Jet Flowfields”.
AIAA Paper 98-2910, June 1998.
0 90 180 270 360
Phase angle (deg)
−30
−20
−10
0
10
20
30
V−
ve
l (m
/se
c)
Exp. data, PIV
tlns3d−sa (baseline)
tlns3d−sa (cg)
tlns3d−sa (fg)
tlns3d−sa (low dt)
tlns3d−sst (baseline)
Figure 1: Time-variation of V-velocity at slot exit, x=0., y=0.1 mm
1.4.4
0 4 8 12 16 20
y (mm)
0
2
4
6
8
10
ce
n
te
rli
ne
 V
−v
el
oc
ity
 (m
/se
c)
Exp. data, PIV
Exp. Data, Hotwire
tlns3d−sa (baseline)
tlns3d−sa (cg)
tlns3d−sa (fg)
tlns3d−sa (low dt)
tlns3d−sst (baseline)
Figure 2: Center-line average velocity comparisons
0 1 2 3 4 5 6 7 8
Jet Width (mm)
0
2
4
6
8
10
12
14
16
18
20
Y 
(m
m)
Exp. Data (PIV)
tlns3d−sa (baseline)
tlns3d−sa (cg)
tlns3d−sa (fg)
tlns3d−sa (low dt)
tlns3d−sst (baseline)
Figure 3: Jet-width comparisons
1.4.5


Simulation of a Synthetic Jet in Quiescent Air Using TLNS3D Flow Code

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