TECHNICAL PAPER
Three-dimensional hydrodynamic model for wind-driven circulation
C. L. N. Cunha1•A. C. Scudelari2•P. C. C. Rosman3Received: 7 March 2016 / Accepted: 23 November 2017 / Published online: 30 January 2018 The Brazilian Society of Mechanical Sciences and Engineering 2018
Abstract
This paper presents the development of a three-dimensional hydrodynamic model for wind-driven circulation using the moving element method in the vertical discretization. As most three-dimensional models use sigma coordinate transfor-mation in the vertical discretization, the use of the moving element method is an alternative to the latter, with the advantage that there is no need for fixed subdivision of the water column. In the proposed model, the coupling of two- and three-dimensional hydrodynamic model is considered. The shallow-water equations are integrated in the vertical direction, finite elements are employed in the spatial discretization, and finite differences in the time discretization, to resolve the position of the free surface (f), and the vertically integrated velocities components. The three-dimensional module is used to compute the velocity profiles. In the three-dimensional module is employed the moving element method in the vertical discretization. The efficiency of the model is demonstrated through the comparison of its results with laboratory experi-mental results and with another model that uses sigma coordinate transformation in the vertical discretization.
Keywords Three-dimensional model Semi-implicit model Shallow-water equation Wind-driven circulation
1 Introduction
Economic and social development in areas close to water bodies has caused growing environmental risk, not only in coastal regions, but over the entire supply system: this imbalance is caused fundamentally by the release of industrial and domestic waste and the disorganized natural resources exploitation in these areas. For this reason, in recent years, the environment preservation has become a relevant factor in promoting social wellbeing.
One of the main supporting tools in environmental management is the numerical models such as hydrody-namic models. The characterization of the hydrodyhydrody-namic circulation of coastal water bodies by numerical modeling is essential to assess and to monitor environmental impacts suffered in these regions; in this way, numerical modeling is becoming an indispensable environmental management tool.
The three-dimensional mathematical models, describing the hydrodynamics circulation of the water bodies, are well known, and are formed by the continuity equation and the Navier–Stokes equations. The numerical methods gener-ally used to solve this system of equations are the finite-difference (FDM), finite-element (FEM), and finite-volume methods (FVM). The FDM [2,10,12,25] has been widely used in the area of fluid flow, applied to uniform grids. A significant advantage of the FEM [14,16,26] is its easiness in describing complex geometries. Recently, the FVM [4,6,9,13] has been successfully applied in computational fluid mechanics because of the easiness of association between physical and mathematical models.
The multilayer model is a three-dimensional model wide-ranging in the literature [15,29], in which the water body is divided into homogeneous layers solved according to a two-dimensional model, with integrated equations
Technical Editor: Andre´ Cavalieri. & C. L. N. Cunha
cynara@ufpr.br
1 Programa de Po´s-Graduac¸a˜o em Engenharia Ambiental,
Universidade Federal do Parana´, Caixa Postal 19011, Curitiba, PR CEP 81531-990, Brazil
2 Programa de Po´s-graduac¸a˜o em Engenharia Sanita´ria,
Universidade Federal do Rio Grande do Norte, Campus Universita´rio s/n, Lagoa Nova, Natal, RN CEP 59072-970, Brazil
3 Programa de Engenharia Oceaˆnica, COPPE, Universidade
Federal do Rio de Janeiro, Caixa Postal 68508, Rio de Janeiro, RJ CEP 21945-970, Brazil https://doi.org/10.1007/s40430-018-0977-z(0123456789().,-volV)(0123456789().,- volV)
along the vertical direction. Kinematic and dynamic boundary conditions are imposed along the interfaces.
Another usually used three-dimensional model is known as the Quasi-3D (Q3D) [10,11,13], in which the equations are simplified by adopting shallow-water conditions, that is, by applying hydrostatic distributions for pressure fields. In this model, vertically integrated shallow-water equations are used to solve the mean velocity field, and for every time step, the velocity profiles are determined at each node of the grid. Another three-dimensional models have been developed, that solve vertical and horizontal modules separately, e.g., [5,17, 19,32]. In these models, the free surface position (elevation) and the components of verti-cally integrated velocities are determined using two-di-mensional equations, forming the two-ditwo-di-mensional module (2DH). In the three-dimensional module, velocity fields are solved based on three-dimensional momentum conserva-tion equaconserva-tions, since only the derivatives of each variable in the vertical direction are implicit. An additional model can also be included to determine the density field, which implies calculating the salinity and/or temperature trans-port equation.
Most the three-dimensional models use sigma coordi-nate transformation in the vertical discretization. The model presented in this study employs the moving element method in the vertical discretization [24] as an alternative to the sigma coordinate transformation, since there is no need for fixed subdivision of the water column.
The moving element method has been proposed in many different contexts over the last decades. However, in most of the applications, the moving element method was used only to solve the bi-dimensional flow. Scudelari [24] used the moving element method for the solution of the bi-dimensional shallow-water equations and compared its results against the analytical solution, using irregular mesh, obtaining a good agreement. Stockstill [27] describes a method for determining implicitly the waterline and flow variables in shallow water. The model uses an implicit Petrov–Galerkin moving-finite-element representation of the two-dimensional shallow-water equations. The model offers a viable means of representing shallow-water flows where the boundary locations are not known a priori. Baines et al. [1] used a moving mesh finite-element algo-rithm for the adaptive solution of non-linear diffusion equations with moving boundaries in one and two dimensions, applied in problems involving time-dependent partial differential equations with moving boundaries. Vanzo et al. [28] derived a novel numerical method to discretize the system of governing equations for the pol-lutant transport by shallow-water flows over non-flat topography and anisotropic diffusion. The resulting method is implemented on unstructured meshes, and is assessed for accuracy, robustness, and efficiency on test
problems including non-flat topography. The novelty of this work is the use of the moving element method in the vertical discretization. The advantage of this method is the versatility of the discretization, once there is no need of a fixed water column division, which allows a better adjustment in the vertical discretization in the water bodies with non-flat topography.
The three-dimensional model, presented in this study, is calculated in two modules: in the 2DH module, the free surface position (f) and the components of vertically integrated velocities (U and V) are calculated using the continuity equation and momentum conservation equations in the x- and y-directions, vertically integrated. The decoupling method employed to solve the two-dimensional model prevents the simultaneous solution of several unknowns in the system of equations [7]. Thus, only the free surface position is implicitly solved, while velocity components are explicitly solved. The three-dimensional model uses three-dimensional momentum equations, in the x- and y-directions, and the three-dimensional continuity equation to calculate velocity profiles. This model applies the moving element method in the equations discretization and does not use sigma coordinate transformation. The technique allows greater versatility in the discretization of water bodies exhibiting large bathymetry variations, cre-ating a vertical discretization that is updated at each time step. This discretization is based on the grid used in the 2DH module. The two modules are coupled using free surface gradients and bottom shear stress. The present study describes only the three-dimensional module. For additional information concerning the two-dimensional module, the reader is referred to Cunha and Rosman [7]. The model was tested against laboratory data for wind-induced currents, developed by Yu [31], and compared with another three-dimensional model that uses sigma coordinate in the vertical discretization [23].
Rosso and Rosman [23] developed a three-dimensional model for the Navier–Stokes equations with shallow-water approximations, considering hydrostatic pressure approxi-mation, named FIST3D. In the FIST3D, the vertical spatial discretization is carried out through finite differences using sigma coordinate transformation, whereas the spatial dis-cretization in the horizontal x–y plane is done through sub-parametric Lagrangian finite elements. The model is composed of two modules: a depth-averaged or 2DH module, through which the free surface elevation and the vertically averaged 2DH current velocities are computed; and a 3D module, that computes the vertical profiles of the horizontal velocity field, as well as the vertical component of the velocity vectors. In FIST3D, the turbulent stresses and turbulent viscosity coefficient parameterizations are identical to the model presented in this paper.
2 Mathematical model
The three-dimensional model is calculated in two modules: first, the depth-averaged shallow-water equations are computed: the free surface position (f) and integrated velocity components in the vertical direction (U and V) are calculated [7]. Then, in the second module, the vertical distribution of velocity components is determined. The basic equations used in this module in Cartesian coordinate system (x, y, z), aligned to the east, north, and vertical directions, are as follows:
Equation of continuity: ou oxþ ov oyþ ow oz ¼ 0: ð1Þ Equations of motion: ou otþ u ou oxþ v ou oyþ w ou oz¼ g of ox þ 1 qo osxx ox þ osxy oy þ osxz oz þ fv ð2Þ ov otþ u ov oxþ v ov oyþ w ov oz¼ g of oy þ 1 qo osyx ox þ osyy oy þ osyz oz fu; ð3Þ where the unknowns are u, v, and w, representing the velocity components in directions x, y, and z; g is gravity; qois the specific reference mass; t is time; sij(i = 1, 2, 3;
j = 1, 2, 3) corresponds to turbulent stresses, and f is the Coriolis factor, which represents horizontal acceleration, denoted by:2Xsenh, where X is the angular velocity of the Earth and h is the latitude of the site considered. In this case, the effects of vertical Coriolis acceleration are dis-regarded. Turbulent stresses can be written in a parametric form by following the approach presented by Rosman and Gobbi [22], which is based on the filtering techniques [21]. These stresses are written as follows:
sT ij qo ¼X 4 k¼1 Kijdjkþ K2k 24 ouj oxk ou i oxk þ Kijdikþ K2k 24 oui oxk ou j oxk ; ð4Þ where i, j = 1, 2, 3 and k = 1, 2, 3, 4 with k = 4 corre-sponding to time t (in this context, x4= t), and Kijare the
turbulent viscosity coefficients. The spatial and temporal Gaussian filters widths, in the xkdimension, are defined as
Kk= akDxk, where akis a homogeneous scaling parameter
in the xkdimension. The value of akcalibrates the filtering
terms magnitude. Usual values of akare between 0.25 and
2.0, and most often good results are obtained with
ak= 1.0. It is easy to verify that the filtering terms, as
displayed in Eq. (4), behave as self-adjusting sub-grid scale turbulent stresses. It is also verified that in a non-structured finite-element discretization mesh, the magnitude of these terms is a function of the local resolvable scale.
2.1 Boundary conditions
When the vertical distribution of velocity components is determined, boundary conditions must be specified at the free surface and the bottom.
2.1.1 Bottom boundary condition
For the boundary condition, bottom velocity is equal to zero, that is:
uijz¼h¼ 0 ði ¼ 1; 2; 3Þ: ð5Þ
Coupling between the three-dimensional and two-di-mensional models is achieved through bottom shear stress. Thus, bottom shear stress is defined as follows:
sF i qo ¼ ffiffiffi g p C j jUu i ði ¼ 1; 2Þ; ð6Þ
where u* is the characteristic friction velocity, obtained
from the profile of velocities calculated in the 3D model, C is the Che`zy coefficient, and Uirepresents the integrated
velocity component in xi-direction, previously calculated in
the two-dimensional model (2DH). 2.1.2 Free surface boundary condition
The boundary condition is the specification of surface stresses, in directions x and y. These stresses can be parameterized as follows:
sSi ¼ qarCDWi102 cosb ði ¼ 1; 2Þ; ð7Þ
where qaris the specific mass of the air, W10represents the
wind velocity, measured 10 m from the free surface, b is the angle of wind direction in relation to the xi-axis, and CD
is the drag coefficient, defined by Wu [30] as follows:
CD¼ 0:001 ð0:8 þ 0:065W10Þ: ð8Þ
2.1.3 Land boundary conditions
For land boundary points exhibiting significant inflows or outflows such as a river or a small estuary ending in a bay, for an instantaneous flow situation, prescribing only the normal velocity component is not sufficient to define a well-posed problem. In fact, when boundary segments have significant inflows, in addition to the normal flux or
velocity component, the tangential flux or velocity along the segment must be specified, and it is set equal to zero.
2.2 Parameterization of the coefficient
of turbulent viscosity
As filtration represents the greatest portion of horizontal momentum diffusion, the horizontal turbulent viscosity coefficient has a little influence. By contrast, vertical tur-bulent viscosity coefficients are more important and essential to the predictive of numerical models capacity. The vertical turbulent viscosity coefficients follow para-bolic distribution and can be defined, according to Fischer et al. [8], as follows: Ki3 ¼ a j uj jiz 1þ z d i¼ 1; 2 ð Þ; ð9Þ
where j is the von Karman constant,
d (x,y,t) = h (x,y) ? f(x,y,t), h(x,y) is the water depth, d(x,y,t) is the total water depth, and a is a calibration parameter.
The characteristic friction velocity can be calculated based on stress at the surface (sS
i) and at the bottom (sFi), or
expressed as a function of mean flow variables. Considered as a function of surface and bottom shear stress, charac-teristic friction velocity can be defined by the expression below: uSi¼ ffiffiffiffiffi sS i qo s or uFi¼ ffiffiffiffiffi sF i qo s ; ð10Þ where uS
icorresponds to the surface characteristic friction velocity and uFi is the bottom characteristic friction velocity.
Bottom characteristic friction velocity is assumed to be a function of the logarithmic velocity profile. In other words, close to the wall, velocity distribution can be well represented by a logarithmic function as follows:
uFi¼ juiðDzÞ
ln32:6Dz2e ; ð11Þ
where ui(Dz) depicts velocity at the first level above the
bottom and e is the length of equivalent bottom roughness. The above formulations show that the characteristic friction velocity can take on several values, which may differ substantially amongst themselves. Jin [11] concluded that the best determination for u* follows the choice
between maximum values of uS i and uF i. Hence: ui¼ max uFi; uSi : ð12Þ
For flows where bathymetry changes abruptly, the near-bottom velocity profile is subject to the effects of recircu-lating currents. Selecting characteristic friction velocity in
accordance with Eq. (12) has a little significance for flow, given that the velocity profile tends to invert close to the bottom. In such cases, another characteristic friction velocity can be calculated, which is based on mean flow variables, by the coupling of the 3D and 2DH modules. Characteristic friction velocity for 2DH is denoted by the following: u2DH i¼ ffiffiffi g p C Ui: ð13Þ
The characteristic friction velocity is then determined according to the expression below:
u¼ maxðuF; uS; u2DH Þ: ð14Þ
As previously mentioned, filtration represents a sub-stantial portion of horizontal momentum diffusion. There-fore, a simplified formulation for the turbulent viscosity coefficients can be adopted, which is obtained from the vertical means of Eq. (10), considering j = 0.4. Thus, the horizontal turbulent viscosity coefficients for the two-di-mensional model can be defined as follows:
Kij¼ KVijþ KHijffi 0:067ud ði; j¼ 1; 2Þ; ð15Þ
where KVij are the diffusion coefficients for conservation of
momentum due to vertical averaging and KHij are the
dif-fusion coefficients for horizontal momentum conservation. The model solves the vertical and horizontal modules separately; the vertical turbulent viscosity coefficients are depicted in Eq. (9) and the horizontal mixing coefficients are shown in Eq. (15).
3 Numerical model
The numerical model uses a combined system: the implicit factored scheme in the time discretization (finite differ-ences), the finite elements in the two-dimensional spatial discretization, and the moving element in the three-di-mensional module spatial discretization. In both the two-and three-dimensional modules, the numerical method uses a decoupling technique, where only one variable is calcu-lated implicitly and the remaining variables are determined explicitly. This decoupling is carried out via the Successive Substitution Method (SSM).
The calculation process employed for the three-dimen-sional model, denominated MOTRID, can be described as follows: the free surface position (f) and the vertically integrated velocities (U and V) are calculated in the 2DH module.
With the free surface position calculated for the entire domain, the three-dimensional velocity components u and v are obtained from the equations of motion (Eqs.2and3) and w, from the continuity equation (Eq.1).
Since the free surface position is calculated using ver-tically integrated velocities, three-dimensional velocity profiles must be corrected to satisfy the equation of con-tinuity, such that the vertically integrated velocities in both modules are equal. With this purpose, initially, the values obtained from the three-dimensional profiles integration are compared with those arising from the two-dimensional module; afterwards, the velocity profile is adjusted so as to produce the same vertically integrated velocity of the two-dimensional module.
3.1 Temporal discretization via implicit factoring
The reference system used in the three-dimensional module is shown in Fig.1, where the origin of the z-axis is con-sidered on the free surface.
In order to uncouple the equations, the velocity com-ponents u and v are explicitly written in the momentum equations as functions of the free surface position (f) by making use of extrapolations and interpolations. The fol-lowing notation is employed in the discretized version of Eqs. (1), (2), and (3), see Scudelari [24]:
(…)?variables at time t ? Dt; (…) variables at time t; (…)-variables at time t - Dt; (…)=variables at time t - 2Dt;
(…)#extrapolated variables at time
t ? Dt;
(…)extrapolated or interpolated at time t ? 1/2Dt
For the extrapolated or interpolated variables, with a second-order scheme, the following quadratic approxima-tions with three time levels are used [20]:
G#¼ 3G 3Gþ G¼ ð16Þ
G¼ 1:875G 1:25Gþ 0:375G¼ ð17Þ
Using the approximations defined in the implicit fac-toring scheme, the time discrete equation of motion in direction x is given by the following:
2u þ u Dt þ u þou oxþ u ou# ox þ v #ou oyþ v ou# oy þ w #ou oz þ wou þ oz ¼ g o f þþ f ox þ 2 ob xx ox þ obxy oy þ o oz b xzþ w xz ouþ oz " # þ 2fv; ð18Þ where: obxx ox ¼ o ox KxxþK 2 x 24 ou ox ou ox þ Kxxþ K2x 24 ou ox ou ox þ K2y 24 ou oy ! ou oy þ K 2 y 24 ou oy ! ou oy þ K2z 24 ou oz ou oz þ K 2 z 24 ou oz ou oz þ K2t 24 ou ot ou ot þ K 2 t 24 ou ot ou ot ui (x,y,z,t) z = -d(x,y,t) z = z xi 0.0
Fig. 1 Reference system used in the three-dimensional module
Fig. 2 Definition of the element for the moving element method, based on the grid used in the 2DH module and the connectivity of elements corresponding to each node
obxy oy ¼ o oy K2x 24 ov ox ou ox þ Kxyþ K2 x 24 ou ox ov ox þ K xyþ K2 y 24 ov oy ! ou oy þ K 2 y 24 ou oy ! ov oy þ K2 z 24 ov oz ou oz þ K 2 z 24 ou oz ov oz þ K2 t 24 ov ot ou ot þ K 2 t 24 ou ot ov ot obxz oz ¼ o oz K2x 24 ow ox ou ox þ Kxzþ K2 x 24 ou ox ow ox þ K2 y 24 ow oy ! ou oy þ K 2 y 24 ou oy ! ow oy þ K2 z 24 ou oz ow oz þ K 2 t 24 ow ot ou ot þ K2t 24 ou ot ow ot wxz¼ Kxzþ K2 z 24 ow oz :
The values of u1 in Eq. (18) are as follows:
uþ 2 Dtþ ou ox þ wou þ oz 2 o oz w xz ouþ oz ¼ gof þ ox g of oxþ 2u Dt u ou# ox v #ou oy v ou# oy w #ou ozþ 2 obxx ox þ obxy oy þ obxz oz ! þ 2fv; ð19Þ and Eq. (19) can be rewritten as follows:
Qxuþþ w ouþ oz 2 o oz w xz ouþ oz ¼ Px; ð20Þ where Qx¼ 2 Dtþ ou ox and Px¼ g ofþ ox g of oxþ 2u Dt u ou# ox v #ou oy v ou# oy w #ou ozþ 2 ob xx ox þ obxy oy þ obxz oz ! þ 2fv:
Similarly, the value of v1is calculated according to the following equation: Qyvþþ w ovþ oy 2 o oz w yz ovþ oz ¼ Py; ð21Þ where: Qy¼ 2 Dtþ ov oy and Py¼ g ofþ oy g of oyþ 2v Dt u #ov ox u ov# ox v ov# oy w #ov ozþ 2 ob yx ox þ obyy oy þ obyz oz ! 2fu:
The value of w1 is determined from the continuity equation. The u1and v1values employed to calculate w1 are not the extrapolated values, but rather those corrected according to the velocity profile, as mentioned before. This calculation is performed as follows:
owþ oz ¼ ouþ ox ovþ oy : ð22Þ
The boundary condition for Eq. (22) is as follows [18]:
wjz¼d¼ 0: ð23Þ
3.2 Spatial discretization
The moving element method, which combines finite-dif-ference discretization with interpolation functions typical of the finite-element method, was adopted for the spatial discretization of the three-dimensional model. The ele-ments, for which interpolation functions are constructed, are defined for each discrete point on the grid, see Fig.2, and their connectivities are based on the two-dimensional grid, see Table1. The selected element is an 8-node hex-ahedron: both the unknowns and geometry are approxi-mated by linear interpolation functions, see Fig.3.
The set of linear interpolation functions for the geometry of the element and the remaining unknowns are denoted as follows [3]:
Table 1 Connectivity of ele-ments corresponding to each node Node Element 1 1 6 7 2 14 8 18 20 10 17 11 21 23 13 25 19 29 30 20 28 22 32 34 24
u1¼ 1 8ð1 nÞð1 gÞð1 sÞ u2¼ 1 8ð1 þ nÞð1 gÞð1 sÞ u3¼ 1 8ð1 þ nÞð1 þ gÞð1 sÞ u4¼ 1 8ð1 nÞð1 þ gÞð1 sÞ u5¼ 1 8ð1 nÞð1 gÞð1 þ sÞ u6¼ 1 8ð1 þ nÞð1 gÞð1 þ sÞ u7¼ 1 8ð1 þ nÞð1 þ gÞð1 þ sÞ u8¼ 1 8ð1 nÞð1 þ gÞð1 þ sÞ : ð24Þ Considering the framework shown in Fig.2, the two-dimensional grid defined on the plane of vectors i and j, where the values of the free surface position f?(x,y,t) are calculated, produces a three-dimensional grid where, for each vertical, the equations of motion and the continuity equation are solved separately.
The derivatives involving horizontal gradients can be defined for any point of the element provided that the global coordinates (xo, yo, zo) and, consequently, the natural
coordinates (no, go, so), are known. Using a generic
vari-able f [xo(n, g, s), yo(n, g, s), zo(n, g, s)], its derivatives can
be calculated as follows: of ox¼ X8 i¼1 fi oui on on oxþ oui og og oxþ oui os os ox ¼ 1 JT j j X8 j¼1 fj oui on X8 i¼1 yi oui og ! X8 i¼1 zi oui os ! " ( X 8 i¼1 yi oui os ! X8 i¼1 zi oui og !# þoui og X8 i¼1 yi oui os ! X8 i¼1 zi oui on ! " X 8 i¼1 yi oui on ! X8 i¼1 zi oui os !# þoui os X8 i¼1 yi oui on ! X8 i¼1 zi oui og ! " X 8 i¼1 yi oui og ! X8 i¼1 zi oui on !#) ð25Þ of oy¼ X8 i¼1 fi oui on on oyþ oui og og oyþ oui os os oy ¼ 1 JT j j X8 j¼1 fj oui on X8 i¼1 xi oui os ! X8 i¼1 zi oui og ! " ( X 8 i¼1 xi oui og ! X8 i¼1 zi oui os !# þoui og X8 i¼1 xi oui on ! X8 i¼1 zi oui os ! " X 8 i¼1 xi oui os ! X8 i¼1 zi oui on !# þ oui os X8 i¼1 xi oui og ! X8 i¼1 zi oui on ! " X 8 i¼1 xi oui on ! X8 i¼1 zi oui og !#) ; ð26Þ where JTis as follows: JT ¼ X8 i¼1 xi oui on ! " X8 i¼1 yi oui og ! X8 i¼1 xi oui os ! þ X 8 i¼1 xi oui og ! X8 i¼1 yi oui os ! X8 i¼1 zi oui on ! þ X 8 i¼1 xi oui os ! X8 i¼1 yi oui on ! X8 i¼1 zi oui og ! X 8 i¼1 xi oui os ! X8 i¼1 yi oui og ! X8 i¼1 zi oui on ! X 8 i¼1 xi oui og ! X8 i¼1 yi oui on ! X8 i¼1 zi oui os ! X 8 i¼1 xi oui on ! X8 i¼1 yi oui os ! X8 i¼1 zi oui og !# : ð27Þ Vertical derivatives are approximated using the finite-difference scheme. The framework depicted in Fig.4 is used to represent the vertical system.
With respect to Eq. (20), Qx,k and Px,k are known
coefficients, obtained from extrapolated/interpolated velocities and values previously calculated in the two-dimensional module, such as the free surface position, f?(x, y, t). By assumingowoz 0 and Kxz¼ mx, one has:
akuþk þ bk ouþk oz o oz m þ k;x ouþk oz ¼ ck; ð28Þ V x y z 2 3 4 7 6 5 8 1 4(-1,1,-1) s 2(1,-1,-1) 6(1,-1,1) 1(-1,-1,-1) 7(1,1,1) 8(-1,1,1) 3(1,1,-1) 5(-1,-1,1) Vp
Fig. 3 Mapping of the hexahedron element of volume V in the standard hexahedron element of volume Vp
where: ak¼ Qx;k 2 bk¼ wk 2 ck¼ Px;k 2 :
The vertical derivates are approximated according to the following: ou oz k ¼uk1 ukþ1 2Dz or ou oz k ¼uk ukþ1 Dz : ð29Þ
Equation (28) can be written as follows: Ak;k1uþk1þ Akkuþk þ Ak;kþ1u þ iþ1 ¼ Bk; ð30Þ in which: Ak;k1¼ bk 2Dz mþx;k1 4Dz2 þ mþx;kþ1 4Dz2 mþk Dz2 Akk ¼ akþ 2 mþx;k Dz2 Ak;kþ1¼ bk 2Dzþ mþx;k1 4Dz2 mþx;kþ1 4Dz2 mþx;k Dz2 Bk¼ ckþ mx;k1 4Dz2 mx;kþ1 4Dz2 þ mx;k Dz2 uk1þ 2 mx;k Dz2 uk þ mx;k1 4Dz2 þ mx;kþ1 4Dz2 þ mx;k Dz2 ukþ1:
The resulting system of equations, which already includes bottom and surface boundary conditions, is solved using the TriDiagonal Matrix Algorithm (TDMA). Note that since this model was developed for tridiagonal matri-ces, some values must be extrapolated.
Similarly, the velocity component in y-direction is cal-culated using the momentum conservation equation at y, while the velocity component in z-direction is determined based on the continuity equation, as previously mentioned. The main advantage of the proposed model is its ver-satility with respect to the vertical discretization. In this model, vertical spacing does not need to be uniform and can vary according to the problem bathymetry, since the variables u?(x, y, z, t), v?(x, y, z, t), and w?(x, y, z, t) are calculated at any point of the z-axis. The sequence used to calculate three-dimensional flow is summarized as follows:
Air intake 3790 39 Ventilator 80 20
Fig. 5 Sketch of water channel used in experiments on wind-driven current (lengths in cm). Adapted from Jin [11]
Fig. 4 Reference system used in the three-dimensional module
Table 2 Parameters of the experiment developed by Yu [31] and presented by Jin [11] Case uw (cm/s) ua(m/s) zo(mm) 1 4.75 - 5.7 0.2570 2 10.4 - 5.7 0.0553 3 14.1 - 5.7 0.0486 4 18.0 - 5.7 0.0491 5 10.4 8.0 0.3790 6 14.1 8.0 0.1340 7 18.0 8.0 0.0187 8 21.4 8.0 0.0268 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 6 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 1
• The free surface position is calculated, in the 2DH module, as a function of vertically integrated variables. • The components of integrated velocities, U? and V?, are calculated along the vertical direction through the 2DH module.
• For each node on the horizontal grid, where the free surface position is calculated, the velocity components in x- and y-directions, u?and v?, are computed along the water column.
• A comparison is made between mean vertical values, determinated from the integration of u? and v?, and 2DH mean velocities, U? and V?, calculated in the second stage. The values of u? and v? are then adjusted, so that the mean vertical velocities, estab-lished by integrating the 3D results, are equal to those of the 2DH;
• The component w?of velocity is directly calculated by the continuity equation, using the corrected values of u?and v?. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 7 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 2
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid SisBaHia
Fig. 8 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 3
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 9 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 4
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 10 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 5
4 Applications
An important step in the development of a numerical model consists in the verification of the model against available data sets, which can be obtained from laboratory experi-ments and from another numerical model already tested. For this model, two test cases were undertaken.
4.1 Wind-induced circulation
The three-dimensional model was validated by comparing results obtained by MOTRID with laboratory data sets. An experimental study of turbulent flows induced by wind was
conducted by Yu [31] and described by Jin [11]. The test objective is the three-dimensional MOTRID code valida-tion, demonstrating its applicability in wind-forced channel analyses. The numerical velocities profiles obtained by MOTRID are compared with the velocities profiles mea-sured from the experiment by Yu [31], and with profiles obtained by Rosso and Rosman [23], for a three-dimen-sional model that uses sigma coordinate in the vertical discretization, FIST3D. The experiment by Yu [31] takes into account 24 combinations of air flow and water velocities for different wind conditions.
The experiment consists of a channel approximately 0.38 m long with 0.20 m water deep, with a cross section of 0.80 m 9 0.59 m. The wind on the surface generates uniform and constant stress along the entire channel. The air current velocity, measured at 0.10 m above the water surface, varies from 5.7 to 8.0 m/s. A pumping system ensures steady flow in the channel, with cross-sectional averaged velocities varying from 4.75 to 21.4 cm/s. During the experiment, wind direction agrees or not with the flow direction. Figure 5 shows a schematic diagram of the experiment.
All tests were carried out in the hydraulically smooth regime. The measuring section, where profiles were determined, was located near the middle of the channel, where the water depth exhibited a little effect caused by the wind set-up on the free surface. Jin [11] describes only the most significant experiments, listed in Table2, showing the mean velocity (uw), the wind velocity measured 0.10 m
above the free surface (ua), and a roughness parameter zo,
depicting equivalent bottom roughness defined as e¼ 15zo:
Numerical simulations were carried out using a grid containing 20 elements, 105 nodes, with constant spacing 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 11 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 6
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 12 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 7
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 U(m/s) 1-z/ D
Yu, 1987 Motrid FIST3D
Fig. 13 Comparison between numerical model predictions (MOTRID and FIST3D) and measured values from Yu [31]—Case 8
of 2.0 m in the x- and y-directions. Forty levels were considered in the vertical direction. At the inlet and outlet of the channel, constant normal velocity was imposed, as well as U = V = f = 0.0 as the initial conditions. The time step used in the simulations was 5.0 s, and the depicted results are related to time 2,000 s. The turbulent viscosity coefficients were parameterized with a equal to 1.0.
Figures6, 7, 8, 9, 10, 11, 12, 13 show that results obtained by the model are very close to the measured values. In cases of opposing wind, numerical profiles fit well to the measured values, both near the bottom and on the surface. In situations where wind direction was the same as that of the current, numerical profiles depart
somewhat from near-bottom values, displaying an excel-lent fit in regions close to the free surface.
The Root-Mean-Square-Error (RMSE) values were determined between prediction and measured values. The RMSE error was found to be 10.26%, for case 1, and 8.46%, for case 2. With higher velocities, RMSE decreased: 2.67%, 3.05%, 3.06%, 5.54%, 2.30%, and 2.88% for cases 3, 4, 5, 6, 7, and 8, respectively. In cases 7 and 8, the higher velocities could generate numerical instability in the model. However, the results show that the MOTRID model is accurate in simulating flows with a flat bottom and subject to wind action. The type of flow rep-resented by this experiment is common in regions subject to wind action, easily found in nature.
Fig. 14 Visualization of the bathymetry of the channel used in the experiment 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 5 10 15 20 25 30 35 40 45 50 Section U (m/s) Section 1 Section 2 Section 3 Section 4 Section 5
Fig. 15 Variation of the integrated vertical velocity component in the x-direction, with the location of the control sections
4.2 Channel with non-flat topography
To evaluate the model performance and to demonstrate its three-dimensional predictive capacity, MOTRID was applied to a non-flat topography channel, with two-di-mensional behaviour in profile (x–z plane) and no gradient in the y-direction. The channel has approximately 960 m long with 30 m wide.
Numerical simulations were carried out using a grid containing 72 elements, 343 nodes, with constant spacing of 3.53 m in the x- and y-directions. Forty levels were considered in the vertical direction. The channel bathy-metry used in the experiment is illustrated in Fig.14. The depth varies from 2.0 m, in Sect.1, to 8.0 m in Sect.3, remaining constant up to Sect.5. The maximum channel slope is 1.87%, varying according to the flow direction. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 . 0 4 . 0 3 . 0 2 . 0 1 . 0 0 u (m/s) 1-z/D Motrid FIST3D
Fig. 16 Comparison between numerical values calculated by MOTRID and by FIST3D— Sect.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 . 0 4 . 0 3 . 0 2 . 0 1 . 0 0 u (m/s) 1-z/D Motrid FIST3D
Fig. 17 Comparison between numerical values calculated by MOTRID and by FIST3D— Sect.2
At the inlet and outlet of the channel, constant normal velocity was imposed. Figure15 shows, in each channel section, the velocity component integrated vertically in the x-direction. The equivalent bottom roughness was 0.0815 m. For the time step equal to 5.0 s, the maximum Courant number is 1.6 and the minimum, 0.4. The initial conditions were U = V = f = 0.0.
The results are shown for five different sections, at time equal to 2000s; the sections are shown in Fig.15. The
velocity gradient is maximum in Sect.2, whereas Sect. 3
presents a slightly lower velocity gradient. The other sec-tions do not present any velocity gradients in the x-direction. Figures16, 17, 18, 19, 20 show a good agreement between MOTRID and FIST3D results; the FIST3D results are assumed as the reference ones. In Sect.1 (Fig.16), where there is no variation in bathymetry, the results obtained are very close and depend only on the local variables of each node of the two-dimensional mesh.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 . 0 4 . 0 3 . 0 2 . 0 1 . 0 0 u (m/s) 1-z/D Motrid FIST3D
Fig. 18 Comparison between numerical values calculated by MOTRID and by FIST3D— Sect.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 . 0 4 . 0 3 . 0 2 . 0 1 . 0 0 u (m/s) 1-z/D Motrid FIST3D
Fig. 19 Comparison between numerical values calculated by MOTRID and by FIST3D— Sect.4
In Sects.2 and 3 (Figs.17 and 18, respectively), the bathymetry variations produce differences in the two for-mulations, mainly near the bottom. The differences can be attributed to the treatment of the advective terms near the bottom which, in these sections, is significantly different for the two models. Sections4, 5 (Figs. 19 and 20, respectively) show the same differences observed in the previous sections.
5 Conclusion
This work presents a three-dimensional hydrodynamic model for wind-driven circulation, with a two-dimensional module, which calculates the free surface position and the vertically integrated velocity fields, and a three-dimen-sional module, which calculates the velocity profiles, using the moving element method.
Coupling of the two modules is performed via the free surface gradient and bottom shear stress. In addition, velocity profiles calculated in the 3D module are adjusted, so that mean vertical velocities are the same in both modules.
In the proposed model, vertical spacing does not need to be uniform and can vary according to the water bodies’ bathymetry, diversely from the models that use the sigma coordinates in the vertical discretization. Therefore, the simulation of water bodies with non-flat topography, or with significant variations of the free surface position, can be done in a more efficient way: as there is no need of a fixed division of the water column, deeper regions can have more calculation points than the shallower ones, with a
better adjustment in the vertical discretization. This is the main advantage of the model. Future studies can be developed to ensure applications in natural water bodies with more complex geometries.
The moving element method, used for spatial dis-cretization of the three-dimensional module, proved to be efficient, accurate, and stable in simulating flow through the flat-bottomed channel subject to wind action and, also, on a channel with non-flat topography. It is important to mention that, for this type of flow, the correct discretization of advective terms is essential for describing velocity profiles.
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