Stochastic CFD numerical model, algorithm, and simulation on regular wave fields

This paper invites higher-order derivative numerical technology to research the stochastic characteristics in regular wave fields. The randomness of the parameters vector was incorporated during the stochastic space establishment, by which, the stochastic CFD numerical model and algorithm were developed in the study. Meantime, Taylor expansion coupled with the perturbation was also utilized to realize the investigation on the random behaviours of the regular wave fields where the perturbative values of the parameters vector included 1%, 2%, and 3%. The results indicated the merit of the stochastic CFD numerical model and algorithm in wave field research.


Introduction
The stochastic characteristics exist in all ocean engineering where the numerical simulation must include the specific algorithm.Wilson et al., offering the validated simulative approach, however, failed to include the stochastics in their work [1].Nowruzi et al. also demonstrated the traditional technology in maritime engineering.Meantime, they did not incorporate the stochastic characteristics of the regular head waves in their study [2].However, there are still few efficient stochastic numerical technologies nowadays.The stochastic space must be adopted to define the corresponding random parameters that are the essential characteristics of ocean engineering cases.Shabani et al. analyzed the elementary characteristics of ocean engineering cases.The random parameters in the goal ocean engineering case, however, were not included in their research [3].The coupled cases also had a close relation with the definition of the random parameters in [4] where the stochastic approach was not the properly designed one.At the same time, the proper ideology is the crucial one that helps establish efficient computation technology.McVicar et al. introduced some ideological techniques to deal with marine engineering cases.However, the applicability of their method was not enough to work out the complicated cases [5].Wang et al. properly investigated the regular wave fields as well as their complexity and offered the applicable numerical technology [6].The higher-order derivative numerical technology will be adopted in this paper to realize the stochastic calculation on the random ocean engineering cases.Furthermore, the applicable program was developed in the work of Wang et al., which, in comparison with others, was more comprehensive in the submarine transportation design and research [7,8].

Numerical methods
The higher-order derivative deduction is the key ideology for numerical technology creation.The crucial stochastic characteristics of the goal ocean engineering cases can be defined based on the firstorder moments and second-order moments with the help of higher-order derivative numerical technology.Furthermore, the reliability and failure probability of the goal ocean engineering structures can be computed efficiently with the higher-order derivative numerical technology.In particular, the non-determinate boundary conditions are also able to be included when the stochastic space is built.The random stream and wave in ocean engineering can be expressed precisely by the stochastic numerical algorithm.Therefore, the non-determinate models in ocean engineering cases are developed comprehensively.The work of Almallah et al. failed to compute the irregular head seas by the properly quantitative approach [9].Similarly, Lavroff et al., discussing the hydro-dynamical effects against the naval vehicle, were unable to establish a precise model of the wave impact loads [10].The study adopted the stochastic ideology and created the applicable numerical approach for the random marine engineering cases.
The stochastic study on the wave fields will approach the stochastic space  as the following expression in Equation (1).Based on the stochastic space  defined as the foregoing, the generalized continuity equation on the incompressible viscous flow can be expressed with the higher-order derivative numerical technology as follows: ( ) ( ) The stochastic space  in this paper included the density of the regular wave fields  , the velocities fields of the horizontal flow, and the vertical one of the regular wave fields u and v.
Based on the determinate results on 2-D wave fields, the stochastic momentum equation be developed by the higher-order derivative numerical technology as Equations ( 3) and (4).Hence, the stochastic expression on the equilibrium equation was reported in Equations ( 5), ( 6) and (7).
where the stochastic hydro-dynamical pressure field is designated by p; the stochastic dynamic viscosity is represented by  ; the stochastic dynamic viscous parameter is  that can be computed as 2 3 The stochastic N-S functions on the steady flow can be built as Equations ( 8) and ( 9) when the body force is not included.
The first-order partial derivative on the stochastic hydro-dynamical pressure field can be removed when the ocean current is steady one.At the same time, the rotation operator  can be built based on the velocities fields of the horizontal flow and the vertical one of the regular wave fields u and v as follows: uv yx   =−  (10) The corresponding stochastic partial derivative operator on  is expressed by Equation (11).
uv yx The stochastic N-S functions, in terms of the incompressible viscous flow, can be deduced as the following Equation (12) which is defined by the unique variable  , namely, the rotation operator.
( ) The stream function  is defined by the study as follows: Therefore, the stochastic coupled functions on  and  are established as the following Equations ( 15) and ( 16) in terms of the steady flow.
Particularly, Equation ( 16) can be reduced as the following Equation (17) when the convective item is removed.
( ) Moreover, Equation ( 17) can be simplified as Equation ( 18) when the case is focused on the steady flow field.
( ) The stochastic characteristics of the ocean engineering cases are the key targets that will be quantified with the help of the stochastic numerical technology developed in the study.Particularly, Taylor expansion will be invited here coupled with the stochastic numerical technology developed in the study, by which the stochastic characteristics on the ocean engineering cases can be computed efficiently.The expectation functions on the regular wave fields are established as follows: V is the variance vector of  v ; p V  is the scalar variance of p ; ':' and '  ' are the product operator and the scalar operator.The non-determinate algorithm above is reported in the flowchart in Figure 1 based on the higherorder derivative numerical technology.

Simulation and results
The stochastic CFD numerical model was expressed in Figure 2 where the intake and outlet directions were defined by the infinite elements that were calculated in the unit channel; the bottom and the lateral wings were defined as the wall conditions with the zero velocity components in direction x; the top was defined as the free surface condition with the expectations on the initial velocity components 0 u =1.0 m/s and 0 v =1.0 m/s.Furthermore, it was supposed that the initial velocity components obeyed the uniform distribution with the perturbative values, i.e., the variation values including 1%, 2%, and 3%.The expectations on the stochastic parameters vector  Based on the stochastic CFD numerical model in Figure 2, the stochastic characteristics of regular wave fields were computed with the help of the stochastic numerical technology developed in the study, and the expectations of the stochastic CFD fields with the stochastic parameters vector perturbation were expressed in Figure 3, Figure 4 and Figure 5.The level on stochastic horizontal flow velocity reported the rising feature before velocity of superficial regular wave 36.5 m/s (Figure 3).Especially, the peaks on stochastic horizontal flow velocity stood uniformly at velocity of superficial regular wave 29 m/s.Meantime, the numerical values increased with the perturbative value accumulation.The distribution of stochastic vertical flow velocity showed the chaotic feature when compared with that of stochastic horizontal flow velocity (Figure 4).The main cause was that the stochastic CFD fields were controlled by velocity of superficial regular wave which produced higher perturbation alongside the fields' depth.At the same time, the vector perturbative level on the distribution of stochastic vertical flow velocity was also higher than that of stochastic horizontal flow velocity.The peaks on stochastic hydro-dynamical pressure came from lower velocity of superficial regular wave in comparison with stochastic horizontal flow velocity and stochastic vertical flow velocity (Figure 5).Meanwhile, the distribution of stochastic hydro-dynamical pressure also showed the chaotic feature.The main cause was that the numerical values of stochastic hydro-dynamical pressure got higher sensitivity to the stochastic parameters vector perturbation.

Discussion
The stochastic numerical expressions are built in the stochastic space based on the non-determinate algorithm and higher-order derivative numerical technology.The stochastic numerical expressions built in the study can be used for complicated CFD subjects.Especially, the cases on the stream and wave in ocean engineering can be calculated precisely with the help of the stochastic numerical expressions established in this paper.
Taylor expansion invited here coupled with stochastic numerical technology helps simulate the random behaviors of the ocean engineering cases when the perturbation of the stochastic parameters included 1%, 2%, and 3%.Correspondingly, the expectations of stochastic CFD fields showed cumulative performance, which indicated the complicated essence of the wave fields.The maximal reactive values of u, v, and p attained 35% in comparison with the certain results from the determinate simulation (Figure 3).

Conclusions
Randomness is the most popular phenomenon in ocean engineering cases.The randomness in the ocean engineering cases includes the fluid stochastics as well as the non-determinants on the field and the boundary conditions.Therefore, the proper methodology of the randomness explanation in engineering cases should be the stochastic approach which can help explore the non-determinate characteristics.This paper developed the stochastic CFD numerical model that can realize the comprehensively random simulation on regular wave fields based on the higher-order derivative numerical technology.Furthermore, the relative technology is helpful for numerical program development by the inspiring algorithm introduced in the study.The stochastic CFD fields showed the exactly sensitivity to the stochastic parameters vector perturbation.
refers to the i th stochastic parameter's probabilistic value.N in Equation (1) designates the dimension of O .Thereby, the stochastic set  can be established by the stochastic subset any candidate i  of which represents the stochastic CFD goal field that will be studied in this paper.n of the stochastic set denotes the dimension of  .
stress field of the goal fluid is represented by x

Figure 1 .
Figure 1.Flowchart of non-determinate algorithm on stochastic CFD numerical model.

Figure 2 .
Figure 2. Stochastic CFD numerical model on regular wave fields.

Figure 3 .
Figure 3. Stochastic horizontal flow velocity distribution from expectations of stochastic CFD fields.

Figure 4 .
Figure 4. Stochastic vertical flow velocity distribution from expectations of stochastic CFD fields.

Figure 5 .
Figure 5. Stochastic hydro-dynamical pressure distribution from expectations of stochastic CFD fields.
adopted.The study investigated the random behaviors of the regular wave fields where the perturbative values of