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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 171
Optimizing Turbulence Generation for Controlling Pressure
Recovery in Submarine Launchways
S.Jordan (Naval Undersea Warfare Center, USA)
ABSTRACT
Excess pressure recovered within submarine hull cavities can lead to unfavorable conditions for safe operation. A
notable example would be the submarine vehicle launchway where the vehicle outlet and pump inlet are in far proximity
along the hull. Inasmuch as the main outlet configuration is cylindrical, this topology permits insertion of periodic wall-
mounted devices for controlling its pressure recovery capability. This control is centered on manifesting a pressure loss
through turbulence ingestion caused by the rib's presence. Thus, the work herein presents a numerical investigation by the
large-eddy simulation (LES) of periodic symmetric ribs with a rib height to rib spacing ratio of 5 in a cylindrical duct. The
salient vortical structures produced by the rib's crests are shown as well as the turbulent statistics that comprise the new
kinetic energy responsible for the pressure loss (or reduction in pressure recovery). A comparison of the predicted frictional
coefficient with the at-sea measurements shows that the frictional coefficient is independent over Reynolds numbers
spanning at least three orders-of-magnitude.
INTRODUCTION
Navy design engineers require strict control over the extent of pressure recovered inside submarine recessed hull
cavities and vehicle launchways to insure successful undersea operations during reconnaissance and defense missions. In
cases where the difference between the hydrodynamic performance of the submarine vehicle shutterway recess with respect
to the pump inlet cavity promotes a reverse flow through the connecting launchway mechanism, an adverse pressure
gradient pre-exists across the vehicle just prior to launch. This pressure difference must be overcome by the pump itself
during start-up. Furthermore, given sufficient pressure difference, an initial breachward movement of the vehicle may
occur that can result in unacceptable launch damage. The following work presents a flexible measure for controlling the
level of pressure recovered in the submarine launchway to avoid the adverse launching conditions just described.
Specifically, this measure involves placement of periodic ribs in the submarine cylindrical shutterway end of the vehicle
launchway that can suitable minimize the level of the pressure recovered (or maximize pressure loss) of the internal flow.
Roughening surfaces with small discrete ribs is thoroughly understood qualitatively for enhancing and controlling
mass and heat transfer rates. This knowledge arose largely from the experimental and numerical investigations that
demonstrated their importance in designing efficient ducts for mechanical systems such as aircraft engines and nuclear
reactors. The ribs themselves are typically mounted periodically along the inner walls of the duct and generate new
turbulent regions within the core streamwise flow. This enhanced turbulent activity greatly improves the mixing and/or
cooling characteristics of the ribbed surface over the straight-walled duct. Generally, the accompanying pressure loss is
viewed as a design penalty of the rib element, but as in the present application this consequence is an effective means for
controlling the pressure gradient within the duct.
The flow characteristics of ribbed-wall ducts in the fully rough regime (independent of Reynolds number) fall
basically under one of two categories (Perry, 1969). Roughen surfaces with s/h>4 are termed “k-type” because the rib
presence disturbs the core flow character (s/h denotes the pitch to height ratio of each rib element). Large-scale vortices
shed from the rib crests whose structure remains intact while convected downstream by the core flow. By contrast, a
roughened surface with s/h ≤4 signifies generation of confined vortical structures oscillating between the ribs with minimal
influence on the core flow characteristics. Unlike the ‘k-type', the mean streamwise outer flow of this ‘d-type' roughness
can be quantified by the logarithmic law (using inner variables) that extends down to the viscous sublayer along the rib
crests. In the literature, one will find
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 172
many empirical relationships for approximating the frictional factor (pressure loss) of rib-roughened walls in the fully
turbulent regime (see Fig. 1 and Table 1 for several of them). Unfortunately, given a specific rib height, the choice of its
pitch for achieving an optimum frictional factor is not easily obtained using these relationships. By examining the origin of
each relationship, one will see that the disparity among them arises principally from the various parameters chosen for each
experiment such as the test section, h/D (or h/Dh) ratio and range of s/h ratios studied. Thus, the collapse of these
relationships onto a unique expression for the frictional coefficient apparently requires redefining the s/h parameter.
Alternatively, the choice of s/h for optimizing the pressure loss in the submarine shutterway must be based on the
experimental measurements of Berger and Hau (1979). Although they did not report an empirical expression for the
frictional coefficient, they produced quantitative local mean mass and heat transfer distributions within a roughened
circular cylinder. They tested ribs 3≤s/h ≤10 which showed that the highest mean mass transfer coefficients were achieved
for s/h=5 over a wide range of Reynolds numbers. Berger and Hau concluded that the ratio s/h=5 achieves favorable heat
and mass transfer distributions because flow reattachment and separation occur simultaneously between subsequent ribs.
Thus, the present paper is concerned with a ‘k-type' roughness having s/h=5 as an effective mechanism for maximizing
the mean static pressure loss (or minimizing the mean pressure recovery) within a cylindrical duct. To ascertain the
dominant turbulent physics that are responsible for producing the static pressure loss and the resulting frictional loss
coefficient, a computation was conducted using the large-eddy simulation (LES) methodology. Since the energy-dominate
scales of the turbulent field primarily attribute to the static pressure loss, LES is well-suited for this purpose. The impetus
of the LES methodology is full resolution of the energy-bearing scales of the turbulent motion while modeling the smaller
scales that tend towards homogeneous and isotropic conditions. The salient turbulent features of the flow are presented in
this paper including its structure and statistical quantities that originate from the rib's presence. Although the present
computations depict flow characteristics for a Reynolds number several orders of magnitude lower than full-scale, the mean
pressure loss is verified by at-sea measurements taken from a full scale prototype test on-board a US Navy submarine.
Figure 1. Empirical Relationships for Determining the Friction Factor for Rib -Roughened Ducts.
Table 1. Parameters Determined for the Relationships in Fig. 1;
β
Author B C
−0.24
Whitehead 0.41 4.50 6.53
−3.75
Webb 0.41 0.95 0.53
−3.75 1−h/Rh
W&M 0.41 0.50
Hann 0.40 3.50 1.43 0.35
Karmon 0.40 1.20 ····· ···· ·········
GOVERNING EQUATIONS AND SOLUTION METHOD
To provide sufficient spatial resolution of the salient turbulent activity in the cylindrical duct, grid point clustering is
necessary around the rib elements. Concurrently, application of a LES formulation to this grid topology requires proper
transformation and filtering of the cylindrical coordinate form of the full-resolution equation system (incompressible
Navier-Stokes and continuity equations) to a curvilinear coordinate famework. The procedure suggested by Jordan (1999)
is followed herein where the first spatial operation is formal transformation of each term in the original equations. The
transformed system appears as
Continuity:
(1)
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 173
Momentum [streamwise (x), radial (r), circumferential (θ)]:
(2a)
(2b)
(2c)
where each term is shown in its non-dimensional conservative form; and
where The coefficients and denote the metrics and the Jacobian of the transformation,
respectively. Filtering the above cylindrical system derives the LES formulation. Thus, the resultant grid-filtered equations
in curvilinear coordinates for cylindrical geometries become
Continuity:
(3)
Momentum:
(4a)
(4b)
(4c)
where the resolvable contravariant velocity components in the convective terms are defined; The
subgrid scale (SGS) stress tensor is defined in contravariant form as According to Jordan (1999), the
metric coefficients are considered as filtered because they are evaluated numerically at discrete points along the curvilinear
lines (denoted by a tilde).
The above LES system was time-advanced by a variant of the fractional-step method (Jordan and Ragab, 1996). This
technique utilizes a semi-staggered discretization molecule that is reformulated in boundary-fitted cylindrical coordinates.
The diffusive derivatives were time-advanced by the Crank-Nicolson scheme to eliminate the high viscous stability
restriction near the rib crests, while the non-linear terms were time-advanced by an explicit Adams-Bashforth scheme.
Spatially, the convective derivatives were approximated by third-order-accurate upwind-biased finite differences with the
diffusive terms discretized using standard second-order-accurate finite volume differences. The pressure-Poisson equation
of the fractional-step procedure was also central differenced to the second order. Additional details of the solution
methodology, along with several test cases, can be found in Jordan and Ragab (1996).
DYNAMIC SUBGRID SCALE MODEL
For the ribs computation, all of the turbulent scales removed by the filter operation were modeled by an eddy viscosity
relationship modified for dynamic computation of the model coefficient (Smagorinsky, 1963), (Germano et al., 1991). The
dynamic coefficient will give the correct asymptotic behavior of the turbulent stresses when approaching the rib walls and
minimize SGS contributions in the low turbulent regions between subsequent ribs. The transformed form of the dynamic
model is expressed as
(5)
where C is the dynamic coefficient and the filtered metric term is defined as
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 174
which transforms the trace of the SGS stress tensor. The turbulent eddy viscosity is defined as where
and is the grid-filter width which is equal to the local grid spacing. The resolvable strain-rate field is
expressed as with each term defined by
(6a)
(6b)
(6c)
where To preserve second-order accuracy in time, the Crank-Nicolson and Adams-Bashforth schemes were
applied to the first and remaining components of the total viscous term, respectively.
MODEL COEFFICIENT
A unique expression for the model coefficient was derived by employing the procedures of Germano et al. (1991),
Lilly (1992) and Jordan and Ragab (1998). The procedure requires test filtering the above governing LES equations for
cylindrical geometries. This third spatial operation produces resolvable tensors similar to those obtained by Jordan and
Ragab (1998) for generalized curvilinear coordinate systems. Specifically, two tensors arise that entail a modified Reynolds
stress
(7)
and a modified Leonard stress
(8)
The second overbar in these definitions indicate the test filter operation. Both tensors and are evaluated in the
computation by explicitly filtering the cylindrical and the contravariant velocity components of the resolved field. The
identity for the Leonard term in this curvilinear coordinate system has an identical form to that originally
derived by Jordan and Ragab (1998). Using the same eddy viscosity relationship, modeling the modified Reynolds stress
tensor becomes
(9)
Substituting this relationship into the above identity along with the expression for the SGS stress field defines the
modified Leonard term in the computational space as
(10a)
(10b)
where the filter width ratio is If we now follow the least-squares minimization procedure of Lilly (1992), the
model coefficient is uniquely given by
(11)
As noted earlier, this expression gives both positive and negative values for the dynamic coefficient through the
product Positive coefficients denote forward scatter of energy from the coarse to the finer turbulent scales whereas
the negative values indicate backscatter or energy transfer locally up the cascade. To insure computational stability, the
negative coefficients were truncated to zero. Thus, in those regions of the flow the truncation error of convective term acted
as the SGS model. Explicit filtering by a box-type filter was performed along the curvilinear lines in the computational
space. The filter itself is identical in form to its Cartesian counterpart but requires transformation of the physical variable
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before and after filtering. In the present application, a filter width ratio of α=2 gave acceptable model performance.
RESULTS AND DISCUSSION
The following section presents and discusses the turbulent statistics of a roughened cylindrical duct by square ribs
placed periodically at s/h=5. According to the space limitations of the submarine launchway under consideration herein, the
maximum permissible height to diameter ratio (h/D) of each rib is 0.1. Thus, the
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 175
pitch to diameter ratio of each rib for the present LES computation is s/D=0.5. Proper resolution of the turbulent vorticity
near the rib walls requires respective boundary stretching of the grid points. Choice of the first field point is difficult to
justify ‘a-priori' because the classic law-of-the-wall profile is not expected along any of the duct walls (expect perhaps near
the rib crests). Typically, the first field point in wall units should be on the order of y+<5. Using this criterion as a
requirement on the instantaneous level, the grid cluster was suitably adjusted before collection of the LES results took place
for evaluating the steady-state turbulent statistics.
Figure 2. Stretched Grid Generated for the Rib-walled Duct Computation; 64x141x401 (θ, r, x directions).
Table 2. Geometry and Steady-state Flow Parameters of the LES Computation.
h/R s ∆ymin y+ Rec
0.23 5h 0.013h 0.75 3310
Figure 3. Isosurfaces of Streamwise Vorticity; (a) Max 2.4, Min 1.6; (b) Max. − 1.2, Min. −2.0
The final grid reached for the cylindrical rib application housed 65x141x401 points in the circumferential (θ), radial
(r), and streamwise directions (x), respectively, as shown in Fig. 2 with simulation parameters listed in Table 2. The grid is
orthogonal, but boundary-fitted to all no-slip walls. First point spacing around the ribs is 0.013 (scaled by h). Along the
cylinder walls in the trough region the first point spacing is 0.021. Highest mean values for y+ were found along the rib
crests and typically were y+= O(1).
Although the instantaneous flow is obviously non-periodic, the computation assumed statistical steady-state
homogeneous characteristics at each subsequent rib and trough section. Thus, periodic boundary conditions were used in
the streamwise as well as the circumferential direction. The grid spacing in the circumferential direction was uniform over 0≤θ≤
2π. The Reynolds number (Re) based on streamwise velocity can not be specifically controlled a-priori. But using the
statistical steady-state results that were time-averaged over T=32, Re=3310 which is based on the rib height and mean
centerline streamwise velocity (Uc ). Numerical stability allowed the computation to be time-advanced at ∆T=0.001;
T=tUc/h.
We begin studying the flow characteristics of the rib-wall duct by illustrating the isosurfaces of the streamwise
vortical structure throughout the ribbed-walled cylinder in Fig. 3. While positive magnitudes are plotted in the lower half of
the duct (Fig. 3a), negative isosurfaces are shown in the upper half (Fig. 3b). Away from the walls near the duct core, both
figures clearly indicate elongated self-similar structures that are periodic and appear to originate near the rib crests. In the
trough regions the same periodicity is apparent, but the structural shape and location of the specific vortices differ in each
region. These same characteristics can be seen in Fig. 4 where the streamwise contours of the circumferential vorticity (ωθ)
are plotted on a plane of constant θ (θ= π/2). This figure clearly indicates that the streamwise vortical structures shown in
the previous figure do indeed originate at the rib crests and require about two periodic lengths before fully convected
radially to the primary duct core. These structures house the maximum and minimum magnitudes of circumferential
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vorticity in the entire duct flow. Between these salient
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 176
streamwise structures, a secondary low-vorticity flow exits, but the origin of these secondary structures is not readily
obvious in Figs. 3 and 4. Lastly, the distribution of circumferential vorticity in both trough regions reveals little similarity
and contributes modestly to the primary vortical flow.
Figure 4. Contours of Circumferential Vorticity; Max 5.0,
Min −6.0, Incr. 1.0.
Figure 5. Contours of Streamwise Vorticity; (a) Rib Mid-
plane, Max 3.2, Min −2.4, Incr. 0.2; (b) 1/2h Plane
Downstream of Rib, Max 4.0, Min −4.0, Incr. 0.4
Instantaneous contours of the streamwise vorticity (ω x) on the first rib mid-plane and on a second plane that is 1/2h
downstream of the first rib are shown in Fig. 5. Positive streamwise vorticity is given by the solid contours whereas the
negative values are plotted as dashed lines. On the mid-plane, a circumferential cluster of streamwise structures is clearly
evident that was vertically convected approximately one rib height since their origin along the previous rib crest. A second
cluster can be observed near the primary core flow. Unlike their lower counterpart, these latter structures only fluctuate in
position while being convected downstream by the dominant streamwise velocity component. The plane downstream of the
first rib (Fig. 5b) shows the same lower cluster of streamwise structures as seen in the previous plane. In addition, this plane
reveals the circumferential cluster of streamwise vorticity that was produced just upstream along the rib crest. Careful look
at their circumferential distribution shows pairs of vortical structures of alternating sign. This observation suggests that
these structures are pairs of counter-rotating streamwise vorticity. Lastly, the streamwise vorticity in the trough region of
this figure is of little consequence. Their magnitudes are comparatively low and do not appear to communicate strongly
with the core flow. Specifically, the streamwise structures that are generated along the rib crest do not enter into the trough
region.
We can explore the above claim of counter-rotating vortical pairs by showing the circumferential distribution of the
near the first rib crest. The instantaneous isosurfaces of magnitude Ω=2 that are
vorticity magnitude
shown in the lower half of the duct in Fig. 6 do indeed confirm this observation. Six pairs of vortical structures are clearly
evident that appear symmetric about the crest streamwise centerline. Moreover, the chief content of vorticity in these
structures originates only at the rib's leading edge.
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Figure 6. Isosurfaces of Streamwise Vorticity; Ω =2.0.
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 177
Figure 7. Contours of the Steady-state Fluctuating Quantities; (a) Streamwise Velocity (vx), (b) Radial Velocity (vr' ), (c)
Kinetic Energy (q2), (d) Reynolds Stress (vx vr ), Pressure Coefficient (Cp ')
Figure 8. Mean Streamlines (T=32)
The duct's turbulent statistics of velocity and pressure scaled by the mean bulk velocity (Ub) are given in Fig. 7. These
root-mean square (rms) contours depict circumferential averages as well as time averages over T=32. They include the
resolved as well as the SGS model contributions. The periodicity of each quantity in the trough and core sections of the
duct justifies implementing periodic boundary conditions in the streamwise direction as well as the total time required to
reach statistical steady-state. Interestingly, the normal fluctuations reach their highest values along narrow bands within the
core flow. Highest Streamwise fluctuations (vx′/Ub)rms peak approximately 2h above the rib crests whereas the radial
component (vr′/Ub)rms attains its maximum about 1/2h into the core flow (Figs. 7a and 7b, respectively). By comparing the
relative rms values of these normal components, the streamwise fluctuations clearly dominate the turbulent activity. This
fact is supported by the scaled kinetic energy distributions shown in Fig. 7c. Contours of the streamwise Reynolds stress
component (u′v′ /Ub2) indicate a much wider band distribution of dominant magnitudes with peak values occurring at
approximately h above the rib crests (Fig. 7d). Unlike the velocity quantities, pressure fluctuations (Cp′) peak locally at the
leading edge of the rib's crest (Fig. 7e). Pockets can also be observed about h above the rib's crest, but their values are less
than 1/3 of peak. Finally, minimal turbulent activity is shown in the trough and centerline regions of the duct, which were
not unexpected characteristics for this type of flow.
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 178
Figure 9. Separated Free-shear Layer Growth in the Duct Trough Region.
Figure 10. Streamwise Mean Velocity Profile at the Rib Mid-Point in Local Wall Units.
Like the turbulent rms quantities, the mean streamlines displayed in Fig. 8 support statistical steady-state similarity
when averaged over time T=32. In the mean, the flow remains attached to the rib crests, which indicates a ‘d-type' flow
behavior in contrast to the ‘k-type' unsteady characteristics as described above. Moreover, this observation suggests that the
rib periodicity s/h=5 is actually within a transition phase from a steady ‘d-type' flow behavior to one that is purely ‘k-type'.
Although the streamlines deflect into the trough region, they do not indicate reattachment. This fact leads to a favorable
pressure gradient over the rib crests, which inhibits separation on this surface. We can explore deeper into these
conclusions by noting that the growth of the separated free-shear layer in the trough region is linear as given in Fig. 9.
Using four inflection points, this layer grows according to ∆r/ ∆x=0.174. Thus, the shear layer grows only 2/3h between
subsequent ribs. A profile of the mean streamwise velocity ( ) in wall units is plotted in Fig. 10 at the rib crest mid-
point along with the law-of-the-wall relationship (using the local inner variables) and Spalding's formula (Spalding, 1961).
Overall, the profile is symbolic of a favorable pressure gradient along the rib crest. Up to y+<4, the rib profile follows the
classic linear sublayer relationship (u+=y+) using the local inner variables. Above this point, two distinct relationships are
suggested that intersect at about =. These latter profiles require further attention, but will not be
investigated in the present discussion.
As emphasized earlier, the argument for placing periodic ribs in the cylindrical duct is to introduce a pressure loss at
the core through turbulent ingestion. Using the mean pressure gradient from the LES results at the duct centerline, a friction
coefficient (f=0.6) was calculated; where L is the length of the duct. This value was subsequently
verified by a full-scale test on-board a US Navy submarine to insure its independence on Reynolds number. Specifically,
three at-sea measurement groupings were recorded that gave consistent friction coefficients of f=0.56+0.04 over
3x106
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 179
normal stress clearly dominates contributions to the kinetic energy of the core flow. Conversely, the pressure fluctuations
attain their maximum near the rib crest leading edge.
Finally, the scaled separated shear layer downstream of the rib grows at a rate of ∆r/∆x=0.174. This growth rate shows
that the flow remains detached within the trough region. Conversely, above the rib crests the streamwise velocity profile in
local wall units indicates a non-separating flow under a favorable pressure gradient. The profile only follows the inner
sublayer relationship for turbulent boundary layers up to y+<4.
ACKNOWLEDGEMENTS
The author and investigator of this work gratefully acknowledges the combined support of the Office of Naval
Research (Dr. L.Patrick Purtell, program officer), the Naval Sea Systems Command (Cdr, R. Schulz, PMS 350) and the
In-house Laboratory Independent Research Program (Mr. R.Philips) at the Naval Undersea Warfare Center.
REFERENCES
Berger, F.P.. and Hau, K.F., (1979), ‘Local Mass/Heat Transfer Distribution on Surfaces Roughened with Small Square Ribs,' Journal of Heat Mass
Transfer, Vol. 22, pp. 1645–1656.
Germano, M., Piomelli, U., Moin, P., and Cabot W.H., (1991), ‘A Dynamic Subgrid-Scale Eddy Viscosity Model,' Physics of Fluids, A. 3, pp. 1760–
1765.
Jordan, S.A., (1999), ‘A Large-Eddy Simulation Methodology in Generalized Curvilinear Coordinates,' Journal of Computational Physics, Vol. 148, pp.
322–340.
Jordan, S.A. and Ragab, S.A., (1996), ‘An Efficient Fractional-Step Technique for Unsteady Three-Dimensional Flows,' Journal of Computational
Physics, Vol. 127, pp. 218–225.
Jordan, S.A. and Ragab, S.A., (1998), “A Large-Eddy Simulation of the Near Wake of a Circular Cylinder,” Journal of Fluids Engineering, Vol. 120, pp.
243–253.
Karman, T.Von, (1930), Nachr. Ges. Wiss. Goett. Math-Phys. Kl., pp. 58–76.
Lilly, D.K., (1992), ‘A Proposed Modification of the Germano Subgrid-Scale Closure Method,' Physics of Fluids, A. 4, pp. 633–635.
Perry, A.E., Schofield, W.H. and Joubert, P.N., (1969), ‘Rough Wall Turbulent Boundary Layers,' Journal of Fluid Mechanics, Vol. 37, pp. 383–413.
Smagorinsky, J., (1963), ‘General Circulation Experiments with the Primitive Equations, I. The Basic Experiment,” Monthy Weather Review, Vol. 91, pp.
99–164.
Wassel, A.T. and Mills, A.F., (1979), ‘Calculation of Variable Property Turbulent Friction in Tubes with Heat Transfer in Rough Pipes,' ASME Journal of
Heat Transfer, Vol. 101, pp. 469–474.
Webb, R.L., Eckert, E.R.G. and Goldstein, R.J. (1971), ‘Heat Transfer and Friction in Tubes with Repeated-Rib Roughness' International Journal of Heat
and Mass Transfer, Vol. 14, pp. 601–618.
Whitehead, A.W., (1976), ‘The Effects of Surface Roughing on Fluid Flow and Heat Transfer,' Ph.D. Thesis, Queen Mary College, University of London.
the authoritative version for attribution.
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OPTIMIZING TURBULENCE GENERATION FOR CONTROLLING PRESSURE RECOVERY IN SUBMARINE LAUNCHWAYS 180
DISCUSSION
I.Celik
West Virginia University, USA
Periodic boundary conditions were used in the streamwise direction. This implies infinite number of ribs. How well
does this assumption conferences with the domain developing flow over a finite number of ribs is questionable. But useful
information can still be deduced from LES because the dominant vortical structures will be periodic, because there are
generated via separation of the flow at the sharp edge of the ribs.
DISCUSSION
S.Cordier
Bassin d'essais des Carnes, France
Could you please explain how a periodic streamwise boudary condition can be used to simulate the
turbulencegeneration or vorticity generation in the real system? Were tests performed with more than one rib present in the
computational domain?
AUTHOR'S REPLY
First of all, let me thank Dr. Celik and Dr. Cordier for their comments and interest in the paper. Both discussions
question the use of periodic boundary conditions in the streamwise direction of the ribbed duct for representing a series of
ribs that ingest new turbulent structures into the core flow. The answer to this question rests on justifying a correct set of
flow conditions at these open boundaries because they depend on the physics outside the flow domain. This dependence is
satisfied when setting these faces as periodic, but the turbulence is assumed to be statistically homogeneous or statistically
periodic. Sufficient separation of these boundaries is critical, where ‘a-priori' knowledge is required of the two-point
correlation length of the periodic streamwise structures. In the case of subsequent ribs, the correlation length is simply the
rib's pitch. But for simpler geometries such as the turbulent channel flow, the channel length must be at least twice the
integral scale of the turbulence in the streamwise direction.
The present LES computation tested the periodicity of the duct flow by simulating two ribs. On the instantaneous
level, periodic vortical structures were computed within the core flow that originated along the leading edge of previous rib
crest. These structures were convected radially towards the duct center over two periodic lengths to reach their streamwise
position. The instantaneous structures within the trough regions, however, showed discernible differences. Thus, the trough
regions sufficiently guided the computation towards statistical steady-state given periodic boundary conditions along the
inlet and outlet faces.
Previous data as well as full-scale measurements taken of the present design show that the flow's memory from non-
periodic inlet conditions occurs only over the first two ribs. This length appears independent of the present range of
Reynolds numbers and scales closure of the separated shear layers that are growing radially towards the duct centerline.
Thus, the present LES computation, which employs streamwise periodic boundary conditions, mimics the actual prototype
design from the third rib and beyond. Direct comparisons of the mean flow data taken from the full-scale prototype and the
present computation proved this point. Given a target centerline pressure loss of a prototype design for the submarine
launchway, these LES results provide the required design length plus two additional ribs.
the authoritative version for attribution.
