=Paper=
{{Paper
|id=Vol-1839/MIT2016-p44
|storemode=property
|title= Computational investigation of turbulent flow impact on non-cohesive soil erosion near foundations of gravity type oil platforms
|pdfUrl=https://ceur-ws.org/Vol-1839/MIT2016-p44.pdf
|volume=Vol-1839
|authors=Alexander Yushkov,Igor Nudner,Konstantin Semenov,Konstantin Ivanov,Nazim Geidarov,Sergey Stukolov,Yury Zakharov
}}
== Computational investigation of turbulent flow impact on non-cohesive soil erosion near foundations of gravity type oil platforms==
https://ceur-ws.org/Vol-1839/MIT2016-p44.pdf
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Computational Investigation of Turbulent Flow
Impact on Non-cohesive Soil Erosion near
Foundations of Gravity Type Oil Platforms
Alexander Yushkov1 , Igor Nudner2,3 , Konstantin Semenov2,4 , Konstantin
Ivanov1 , Nazim Geidarov1 , Sergey Stukolov1 , and Yury Zakharov1
1
Kemerovo State University, Krasnaya, 6, Kemerovo, Russia
2
23 State Marine Design Institute Branch of β31 State Design Institute of Special
Constructionβ, St.-Petersburg, Russia
3
Baltic State Technical University, St.-Petersburg, Russia
4
Saint-Petersburg State Polytechnical University, St.-Petersburg, Russia
zaxarovyn@rambler.ru
Abstract. The flow turbulence impact on the formation of erosion areas
near gravity type oil platforms is studied. The SST (shear-stress trans-
port) turbulence model describing large-scale structures in the internal
area and small-scale turbulence in the external area is used for computing
turbulent fluid flow. The model grounded on estimation of turbulent be-
havior of the fluid flow in the bottom flow region where the soil particles
transfer is influenced by fluid flow and sea-bed irregularities is applied
for estimation of soil erosion. Three sets of numerical studies referred to
increase of hydrodynamic values and flow turbulent transition are given.
Keywords: Viscous incompressible fluid, Navier-Stokes equations, non-
cohesive soil erosion, three-dimensional flow, turbulence, gravity type oil
platforms, numerical and laboratory-based experiments
1 Introduction
The application of gravity type oil platforms at shallow water marine coastal
areas is one of the most current means for oil extraction. Processes of sea floor
erosion near foundations of such oil platforms and its stability issues are of great
interest. In the recent years different investigations of those issues were actively
undertaken, by means of laboratory-based and seminatural experiments as well
as by means of mathematical simulation [1], [2]. The papers [2], [3] contain results
of a great number of experimental and numerical studies of non-cohesive soil ero-
sion near the foundation of the Prirazlomnaya platform, comparison charts of
laboratory and simulation experiments, analysis of the impact of different wave
conditions of fluid flow on the process of particles shift of seabed material. In
those papers the laminar model of fluid flow was applied for numerical evalua-
tion of hydrodynamic quantities. The results of the studies [2] show that when
the fluid flow is slow and there are no surface waves, the laminar model pro-
vides a good match with laboratory results (up to 10-15% accuracy) on the one
524
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
hand, and a significant economy of computational resources, on the other hand.
The paper [3] illustrates the waves impact on the structures of soil erosion near
the platform foundation. The estimations executed at different wave conditions
attest the significant change in the pattern of fluid flow and in the structure
of soil erosion when internal flow velocity and wave amplitude are increased.
Under such modes the applied in [3] laminar model becomes inoperative. As
the results of the former studies show, the increase of the hydrodynamic behav-
ior (that corresponds to small surface waves in a natural experiment) leads to
flow laminar-turbulent transition. It is evident that the analysis of such flows
demands application of the valid turbulence models. The current paper stud-
ies the impact of the developed flow turbulence on the formation of outwashes
and inwashes (accretion) near the gravity type oil platforms, both numerically
and experimentally. The π β π, SST (shear-stress transport) turbulence model
is applied for estimation of bottom fluid flow velocity that directly constitutes
the certain structures of soil erosion. For evaluation of soil erosion we apply the
model, which is grounded on the estimation of turbulent properties of fluid flow
in the bottom region of the flow. The results of three sets of numerical and ex-
perimental studies referred to the increase of the hydrodynamic behavior and
flow turbulent transition are given.
2 Flow Model
Variable flow of viscous incompressible fluid with constant properties is described
by three-dimensional system of Reynolds-averaged Navier-Stokes in accordance
with Boussinesq hypothesis on Newtonian turbulent fluid flow where turbulent
viscous stresses are related to average flow properties with the same correspon-
dence as molecular resilient friction is related to velocity field, yet turbulent
viscosity needs to be defined instead of 6 components of Reynolds stress sym-
metric tensor (ππ‘ ):
{οΈ πππ
ππ₯π = 0, (οΈ )οΈ
πππ πππ 1 ππ π πππ (1)
ππ‘ + ππ ππ₯π = β π ππ₯π + ππ₯π (π + ππ‘ ) ππ₯π ,
where ππ - components of velocity vector, π‘ - time, π₯π - Cartesian coordinates,
π - fluid density, π - pressure, π - kinematic fluid viscosity, ππ‘ - turbulent eddy
viscosity estimated on the ground of the applied turbulence model.
The present paper applies π β π , SST (shear-stress transport) turbulence
model [4], where π - turbulent kinetic energy, π - specific dissipation rates referred
to isotropic dissipation as follows as: π = π½ * ππ, where π½ * = 0.09.
The classic π β π model [5] has issues while calculating stream flows due
to extreme sensitivity to boundary conditions in the external flow. The model
π β π, SST, suggested by Menter, eliminates the said imperfection and combines
the benefits of π β π and π β π turbulence models. Menters model applies the
modified π β π model designed for description of large-scale structures in the
internal area, and in the external area - π β π , aimed to solution of small-scale
525
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
turbulence. In Menters model, the model π β π is re-formulated in terms π and
π, and a special transition function is added in the received equations in order
to transfer from one model to another. The model π βπ, SST is grounded on two
equations, the one is for turbulent kinetic energy and the second is for specific
dissipation:
(οΈ )οΈ
ππ ππ * π ππ
+ ππ = ππ β π½ ππ + (π + ππ ππ‘ ) , (2)
ππ‘ ππ₯π ππ₯π ππ₯π
(οΈ )οΈ
ππ ππ πΌ πππ 2 π ππ
ππ‘ + ππ ππ₯π = πππ‘ πππ ππ₯π β π½π + ππ₯π (π + ππ ππ‘ ) ππ₯π
+
(3)
+2 (1 β πΉ1 ) ππ2 π1 ππ₯
ππ ππ
π ππ₯π
,
where (οΈ )οΈ
2 πππ 2
πππ = πππ‘ 2πππ β πΏππ β πππΏππ ,
3 ππ₯π 3
(οΈ )οΈ
1 πππ πππ
πππ = + .
2 ππ₯π ππ₯π
Outlet
Top
Side
Side
Z
Pile
Y
X
Bottom
Inlet
Fig. 1. Flow region boundaries
The transition function πΉ1 is determined as follows as:
β‘(οΈ [οΈ (οΈ β )οΈ ]οΈ)οΈ4 β€
π 500π 4πππ2 π
πΉ1 = tanh β£ min max , , β¦, (4)
π½ * ππ¦ π¦ 2 π πΆπ·ππ π¦ 2
(οΈ )οΈ
β10
where πΆπ·ππ = max 2πππ2 π1 ππ₯ ππ ππ
π ππ₯π
, 10 , π¦ - distance to the surface. Tur-
bulent eddy viscosity is determined by the following formula:
π1 π
ππ‘ = , (5)
max (π1 π, ππΉ2 )
526
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
where βοΈ
π = 2πππ πππ ,
β‘[οΈ (οΈ β )οΈ]οΈ2 β€
2 π 500π
πΉ2 = tanh β£ max , β¦,
π½ * ππ¦ π¦ 2 π
(οΈ (οΈ )οΈ )οΈ
πππ πππ πππ *
ππ = min πππ‘ + , 10π½ πππ ,
ππ₯π ππ₯π ππ₯π
.
ππ = πΉ1 ππ1 + (1 β πΉ1 ) ππ2 ,
ππ = πΉ1 ππ1 + (1 β πΉ1 ) ππ2 ,
πΌ = πΉ1 πΌ1 + (1 β πΉ1 ) πΌ2 ,
π½ = πΉ1 π½1 + (1 β πΉ1 ) π½2 .
The models invariables are: πΌ1 = 59 , πΌ2 = 0.44, π½1 = 40 3
, π½2 = 0.0828, ππ1 =
0.85, ππ2 = 1, ππ1 = 0.5, ππ2 = 0.856.
For completion of the objective it is necessary to postulate the boundary
conditions for the components of velocity vector ππ , pressure π, turbulent kinetic
energy π and specific dissipation π.
In the inlet (Fig. 1): π1 = π’0 , π2 = 0, π3 = 0, ππ ππ = 0, π = π0 , π = π0 ,
where π’0 , π0 , π0 are selected on the basis of comparison of velocity profile at
the entrance with the experimental data. In the outlet: ππ ππ
ππ = 0, π = 0, ππ =
π
ππ π
0, ππ = 0, On the top surface: ππ = 0 for all variables. On the side surface,
pile and bottom: ππ = 0, ππ 6π
ππ = 0, π = 0, π = 10 π½π¦ 2 , where π½ = 0.075 , and π¦ -
distance from the surface to the center of the closest mesh.
3 Scour Model
The paper [6] applies the model of soil erosion [7], where computations are based
on the values of shear stress on the bottom surface. Here bottom is represented
by a certain surface divided into meshes by the grid. At first, the vector of soil
particles transport through each mesh is estimated, and then the height of the
bottom is determined on the ground of mass balance equation.
Within the frames of the present numerical model, particle motion is influ-
enced by the impact of fluid flow and irregularities in bottom surface. For each
design moment, bottom shear stress is estimated at first, and then the calcu-
lation of the particle motion on the bottom surface is done and mass balance
equation is solved.
Two-dimensional coordinate system (π₯1 , π₯2 ) is implemented for the bottom
surface. In order to estimate the stress vector πΒ―π it is necessary to find the product
of stress tensor π and surface normal π Β― , and then the received vector is projected
to the bottom surface.
To determine the vector component πΒ―π it is necessary to know the value of
turbulent eddy viscosity ππ‘ . In case of application of the laminar model of the
527
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
flow while computing velocity field, this parameter was assigned manually to
be equal to the average value ππ‘ at the turbulent flow mode with analogous
parameters (size of order 10β5 β 10β4 ).
After computing the component of shear stress vector, we determine vector
πΒ― = (π1 , π2 ) of bottom seabed material in unit time per unit of length, π-th
component of which is evaluated by the following formula
ππ πβ
ππ = π0 Β· β πΆ Β· π0 Β· , π = 1, 2 (6)
|Β―
ππ | ππ₯π
Here, the first term represents the component of soil particles transport induced
by flow in the basin, and the second addend soil shift due to bottom surface
irregularities.
The accumulation factor is
{οΈ βοΈ β
12 Β· π Β· (π β 1) Β· π3 Β· (π β ππ ) Β· π, π > ππ
π0 = (7)
0, πππ π
the value of soil transport on the horizontal bottom which is equal to zero in case
when Shieldsβ parameter does not exceed the critical value of ππ . Herewith ππ =
π0 sin(πΌ+π)
sin(π) , where π0 - Shieldsβ rejection number for horizontal bottom, πΌ - tilting
angle of bottom, and π - angle of repose equal to 23β . In accordance with this
formula, Shields rejection number compared with this parameter for horizontal
bottom increases while moving up the slope and decreases when moving down.
According to calculations, the value π0 is in the range of 0, 01 < ππ < 0, 06.
After receiving the bed load transfer vector for computing the bottom changes
formed due to soil transfer, the mass balance equation is solved
πβ 1 βοΈ πππ
= Β· , π = 1, 2, (8)
ππ‘ 1 β p π ππ₯π
written relating to function h of the elevation of the formed bottom profile over
its initial level; here π bed load porosity.
4 Results of Numerical and Experimental Studies
The present paper represents several sets of numerical experiments compared
with the laboratory tests laid out in [1] in detail.
Numerical Domain (Fig. 1) corresponds to the operating area in a model
tank with a gravity type oil platform model. The operating area has the size of
12x6 meters. Left bottom corner is considered to be the origin of coordinates.
At a distance of 4 meters from the Inlet there is an oil platform model with the
size of 2x2 meters, the model has chamfered corners (45 degrees) with the sides
of 0.2x0.2 meters. The depth of fluid in the tank is 0.3 meters.
The patterned finite-volume grid consisting of octagons is used for discretiza-
tion of a numerical domain. The partition along axes Ox and Oy amounts for
528
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Height
0.06
0.05
1 0.04
0.03
0.02
0.01
0
Y -0.01
-0.02
-0.03
-0.04
0.5 1
-0.05
2
-0.06
4 -0.07
5
3
0
0 0.5 1 1.5
X
Fig. 2. Structure of soil erosion. Laminar flow model.
20 points per 1 meter, and along axis Oz - 20 points per 0.3 meter with the
concentration to the bottom of a numerical domain (proportion of the vertical
dimension of a near-bottom mesh to the top one 1:60). The vertical dimension
of a near-bottom mesh is 1 mm. Such partition is prescribed by 556 664 points
and it contains 531 200 meshes.
The first set of computation refers to the use of a laminar fluid flow model
studied in [3] in detail. The velocity of inlet flow corresponds to the average by
depth of fluid flow velocity in a laboratory test and is equal to 0.25 m/s. Fig. 2,
Fig. 3 contain soil discontinuity structures received in numerical and laboratory
studies accordingly.
Number of point Numerical result, mm Measurement result, mm
1 -77.75 -75.34
2 74.64 71.81
3 31.39 33.09
4 -34.42 -31.19
5 10.21 11.17
Table 1. Comparison between experimental and numerical studies
The detailed numerical studies of soil erosion undertaken in [3] with the use
of finite difference methods and reproduced in this study with the help of finite
volume approaches show that in case of low velocity (up to 0.3 m/s) the use
of laminar fluid flow model allows to receive good quantitative agreement with
laboratory tests data with the significant economy of computational resources.
529
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
Fig. 3. Structure of soil erosion. Laboratory experiments data (mm).
Table 1 contains the bottom height function value resulted from numeric com-
putation and experiment in the points shown in Fig. 2.
The second set of computation refers to the use of a turbulent fluid flow model
represented in this study, with the same value (0.25 m/s) of inlet flow velocity.
These computations applied structured computational grid. The segmentation
along axes OX and OY is 40 points per 1 meter and along the axis OZ is 40 points
per 0.3 meter, with the concentration to the bottom of the computation domain.
The vertical dimension of a near-bottom mesh is 0.01 mm which corresponds to
dimensionless distance to the wall π¦ + = 1.
Fig. 4 shows streamlines drawn up in horizontal plane with z=0.1 for different
time points demonstrating eddy formation (a), its development (b), separation
(c) and cycle mode (d, e). These figures allow suggesting the periodic nature of
flow. Though the chart (represented in Fig. 5) of correspondence of velocity vec-
tor magnitude to time (0 sec < t < 1000 sec) in the point beside the streamlined
platform with the following coordinates (7, 4.5, 0.15) allows to conclude that the
flow nature is close to periodic, but it is not such.
Fig. 6 contains space patterns of fluid flow demonstrating formation and
development of complex eddies.
Fig. 7 represents the structure of soil erosion received by a numerical exper-
iment.
The results of computation correspond qualitatively and quantitatively with
the data of laboratory studies and results of numerical computation received
with the use of a laminar fluid flow model.
The third set of computation refers to the increase of inlet fluid flow velocity.
Fig. 8 demonstrates the pattern of soil erosion when the inlet flow velocity is
equal to 0.5 m/s.
In this case the flow structure acquires significantly turbulent nature that
stipulates essentially different arrangement of erosion areas and accretion areas.
530
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
6
5
Y
4
3
2
1
0
0 2 4 6 8 10 12
X
a) t=10 s.
Y 6
4
2
0
0 2 4 6 8 10 12
X
b) t = 30 s.
6
Y
4
2
0
0 2 4 6 8 10 12
X
c) t = 80 s.
6
Y
4
2
0
0 2 4 6 8 10 12
X
d) t = 130 s.
6
Y
4
2
0
0 2 4 6 8 10 12
X
e) t = 140 s.
Fig. 4. Streamlines drawn up in horizontal plane z=10.
531
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
0.7
0.6
0.5
U
0.4
0.3
0.2
0 200 400 600 800 1000
t
Fig. 5. Flow time variations in the point (7, 4.5, 0.15)
Z
X Y
a)
Z
Y
X
b)
Z
X
Y
c)
Fig. 6. Formation and development of complex eddies.
532
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
5 Height
0.006
4.5 0.005
0.004
4 0.003
0.002
3.5 0.001
0
Y 3 -0.001
-0.002
2.5
-0.003
-0.004
2
-0.005
-0.006
1.5
1
3 4 5 6 7
X
Fig. 7. Structure of soil erosion. Turbulent flow model.
Height
4.5 0.14
0.12
4 0.08
0.06
0.02
3.5
0
-0.02
Y 3
-0.06
-0.08
2.5
-0.1
-0.14
2
-0.16
1.5
3 3.5 4 4.5 5 5.5 6 6.5 7
X
Fig. 8. Structure of soil erosion. Turbulent flow model in case of high velocity.
5 Conclusion
The present paper represents the numerical studies of the process of soil erosion
near foundations of gravity type oil platforms under conditions of developing
turbulence of fluid flow. The calculation data show that when the flow hydro-
dynamic behavior is of low values, the small-scale turbulence has little impact
on erosion structure, and in this case the laminar flow model for computing
bottom velocity is more preferable. If the input flow velocity is increased, the
turbulence mode of fluid flow obtains the paramount importance for outwash
533
Mathematical and Information Technologies, MIT-2016 β Mathematical modeling
and inwash (accretion) areas, and the application of the turbulent model for
estimating hydrodynamic values becomes principal in this case.
Acknowledgments. The study has been executed within the frames of Federal
Assignment # 1.630.2014/K βSimulation of Fluid Flow with Variable Density
and Viscosity for Applied Issuesβ.
References
1. Zakharov, Y.N., Nudner, I.S., Gaydarov, N.A., Ivanov, K.S., Semenov, K.K., Lebe-
dev, V.V., Belyaev N.D., Mishina A.V., Schemelinin L.G.: Numerical and Experi-
mental Studies of Soil Scour Caused by Currents near Foundations of Gravity-Type
Platforms. In: International Conference on Civil Engineering, Energy and Environ-
ment (CEEE-2014), pp. 190β197. Hong Kong (2014)
2. Zakharov, Y.N., Nudner, I.S., Gaydarov, N.A., Ivanov, K.S., Semenov, K.K., Lebe-
dev, V.V., Mishina A.V., Schemelinin L.G.: Numerical and Experimental Studies
of Soil Scour near Foundations of Platforms. In: Advanced technologies of hydroa-
coustics and hydrophysics, pp. 239β241. SPb (2014)
3. Zakharov, Y.N., Nudner, I.S., Gaydarov, N.A., Ivanov, K.S., Semenov, K.K., Lebe-
dev, V.V., Belyaev N.D., Mishina A.V., Schemelinin L.G.: Impact of waves and
currents on the soil near gravity-type offshore platform foundation: numerical and
experimental studies. In: Twenty-fifth (2015) International Ocean and Polar Engi-
neering Conference ISOPE-2015, pp. 807β814. Kona, Hawaii, USA (2015)
4. Menter, F. R.: Zonal two equation k turbulence models for aerodynamic flows. In:
AIAA 24th Fluid Dynamic Conference. Orlando, Florida (1993)
5. Wilcox D.C.: Formulation of the π β π turbulence model revisited. J. AIAA, 46, No.
11, 2823β2838 (2008)
6. Solberg, T., Hjertager, B. H., Bove, S.: CFD modelling of scour around offshore
wind turbines in areas with strong currents. Technical report 5, Esbjerg Institute
of Technologym Aalborg University (2005)
7. BrΓΈrs, B.: Numerical modeling of flow and scour at pipelines. J. Hydraul. Eng. 125,
511β523 (1999)
534