Study on the Evolution Pattern of Complex Topographic Flow Field in Typical Tableland and Hogback Ridge

The flow field in complex mountainous wind farms is highly intricate, affecting the accuracy of wind resource assessments and the reliability of wind turbine operation. In order to study the evolving characteristics of the flow field in typical complex terrains under different inflow wind directions and wind speeds, this paper, based on the open-source software OpenFOAM, adopts a neutral atmospheric boundary layer model combined with Reynolds-averaged turbulence model to construct a high-precision flow field model for complex terrains. The research reveals that this model requires fewer computational resources and can effectively simulate the flow characteristics around complex terrains. Wind resource distribution in tableland terrains and hogback ridge terrains is more influenced by the height of the mountains, and the recovery speed of the wake behind the mountains decreases with a reduction in slope. In hogback ridge terrains, under a wind direction of 0 °, the wind acceleration effect at the top of the hogback ridge is stronger than under a wind direction of 270°, as it experiences more intense compression from the mountains. This paper can provide reference for micro-siting in complex terrain.


Introduction
Low wind speed wind farms in our country are mostly located in areas with complex terrain.Air flow over complex terrain can cause complex flow separation and convergence [1].Therefore, in-depth study of the flow distribution of typical complex terrain such as isolated hills and terraces is of great significance for the micro-siting of wind farms.At present, research on complex terrain mainly revolves around wind tunnel experiments and numerical simulations.Kondo used an anemometer to measure the wind speed distribution of several groups of two-dimensional slopes with different slopes and studied the wind acceleration effect of slope terrain [2].Bowen conducted wind tunnel experiments on four different slope 2D ridge terrains and found that the turbulence characteristics in the wake region of the mountain changed significantly [3].The turbulence intensity decreased with decreasing slope, and low wind speed areas were prone to higher velocity standard deviation.Ferreira and others studied the flow distribution around 2D sinusoidal hills using both experimental and numerical simulation methods [4].The results showed that the length of the recirculation zone strongly depends on the shape of the hill.Kim and others provided comprehensive results of mountainous terrain flow obtained through experiments and numerical calculations, including a comparison of the flow field between isolated hills and continuous double hills, and the influence of adjacent hills under non-separation flow conditions 2 [5].It was found that low hills do not change the average velocity profile at the top of downstream high hills, but the opposite is true.Liang Sichao and others selected a turbulence model with wall functions and appropriately adjusted the model parameters to simulate the flow around the Askervein mountain [6].After comparing with measured data, the rationality of the improved model was verified.Li Min used the open-source software OpenFOAM to simulate the flow field of three different slope hills mentioned in reference under different inflow conditions [7][8].
The velocity distribution of different slope terrains was studied.It was found that under turbulent inflow conditions, only the dX1 terrain experienced flow separation at the windward slope waist, and under shear inflow conditions, both dX1 and dX2 terrains experienced flow separation at the windward slope foot, while dX3 with a smaller slope did not produce separation.Shi Junjie, Lei Jiao, and others used RANS turbulence model to analyze the distribution of wind speed, turbulence intensity, and wind acceleration factor in the flow field of typical complex terrain such as hills, flat-cut hilltops, cliffs, and high-low hilltops [9][10][11].They obtained the flow characteristics of different terrains and made comparisons.Aiming at the problem of flow field in complex terrain, this paper focuses on the littleattended typical complex terrain of hogback ridges and tableland terrain, and studies the effects of slope and height on the distribution of flow field in complex terrain and the characteristics of the flow field distribution under different incoming wind speeds, starting from terrain modeling.Tian conducted experiments to measure the wake of five wind turbines arranged on 2D slopes with slopes of 0.25 and 0.5 [12].The results showed that the wind turbines on the slopes obtained more wind energy due to the wind acceleration effect, the wake recovery speed increased, and the fatigue load decreased.Ferreira combined experimental and numerical simulation methods to study the flow distribution around 2D ridges [13].The results showed that the length of the wake recirculation zone depends on the shape of the ridge.

Mass Conservation Equations.
The mass conservation equation, must satisfy the law of mass conservation and is written in differential form as follows: () Where Sm in the equation is the mass source term, which represents the mass flow from the diffuse secondary phase to the continuum primary phase supply in the multiphase flow, or is customized by the user..The core concept of the momentum equation lies in Newton's second law : the momentum's rate of change within a micrometric body over time equals the cumulative effect of external forces acting on the micrometric body.In accordance with this law, the equation for the conservation of momentum in an inertial coordinate system is derived:

Conservation of Momentum Equation
where p is the static pressure,  is the stress tensor, g  and F are the gravitational volume force and other additional volume forces, respectively.F can contain other source terms, like infiltration models or defined models by users.In wind farm wake calculations, F is usually set as a drag source term for the convection field of the wind turbine and nacelle.The stress tensor  is expressed as follows: Where  is the molecular viscosity, I is the unit tensor, and the second term on the right represents volume expansion effect.
2.1.3.Numerical Simulation.In this paper, the RANS simulation method is used, in which the solution variables in the transient Reynolds equation are decomposed into time-averaged and fluctuating quantities during the Reynolds averaging process, e.g., the velocity component can be written as: where i u and i u denote the mean and fluctuation values of the velocity component, respectively (i =1,2,3).The equations above are rewritten in Reynolds time-averaged form as follows: ( ) The above two equations are called Reynolds time-averaged NS (RANS) equations.As opposed to the transient NS equation, the extra phases in the equation represent turbulence effects.

Turbulence Model k- Turbulence Model
The k- turbulence models comprises three variants with similar basic form,the primary difference is that: ( The standard k- turbulence model is constructed upon transport equations governing turbulent kinetic energy k and its dissipation rate.Since the model is based on the assumption that turbulence is fully developed and that molecular viscous effects can be neglected, it is only applicable to flows where turbulence is fully developed.
The transport equations are as follows: ( ) ( ) where The turbulent kinetic energy generation rate k P : Under the Boussinesq hypothesis, there is: where S is the modulus of the average strain rate tensor ij S .The two kinds are defined respectively as The turbulent viscosity can be expressed as: The model parameters are: 12 1.44, 1.92, 0.09, 1.0, 1.

Boundary Conditions
In this paper, neutral atmosphere boundary conditions are used with an inlet boundary: where * u is the friction velocity; is the standard deviation of the wind speed, ( ) z is the atmospheric roughness length, indicating the height at which the near-surface wind speed decreases downward to zero;  is the von Karmen constant speed generally taken as 0.4187; z is the height above the ground; C  is the value generally taken as 0.033 for ABL conditions.
The outlet is set to free outflow: (17) The two side faces are set up as symmetric boundary conditions, i.e: (18) The bottom surface uses a wall function, i.e: The value of E is 9.8; Re is the Reynolds number; p y + is the dimensionless distance from a point in the flow field to the wall; y  + is the critical y + for the viscous substrate; 0 " " is the initial value;  is the kinematic viscosity; t  is the vortex viscosity coefficient.

Mountain Dimensions
Complex terrain in this paper are used modeling software spaceclaim combination of stretching to form stl entity, piggyback ridge and tableland terrain modeling s schematic shown in Figure 1 and Figure 2:   There is little difference in the computing domain size and various typical complex terrains grid division, taking the tableland terrain as an example: taking the boundary line of the upstream and downstream of the square tableland terrain, 10D for the downstream, 6D for the upstream, and 1500m for the vertical direction, the grid resolutions in the horizontal and vertical directions are 40, 40, and 20 respectively, the infill area is the surface of the tableland, the infill level is 3, and the total grid number of the platform terrain with a slope height of 50 is about 2 million.The calculation domain and meshing of other terrains are adjusted according to the working conditions, and will not be repeated in this article.

Topography Flow Field Analysis of the Tableland
Figure 3 shows the distribution of wind resources at 120 m above the ground on a tableland with a slope height of 200 and a slope of 30 under the condition of an inflow wind speed of 4.5 m/s.The airflow is accelerated by the climb along the surface of the hill due to the compression of the terrain, but the maximum values of speed occurs at the front and rear of the platform due to the smooth plane at the platform top.Figure 4 shows the wind speed variation at 120 m above the ground in the vertical profile of y=0.Table 2 shows the maximum and minimum values and occurrence locations of wind speed in the upwind and downwind directions under each working condition.
When the inflow wind speed is changed by the same topography, the maximum wind speed and the minimum wind speed appear in the same position.Under the same inflow condition, taking the slope height of 50m as an example, the downwind wind speed of the small slope terrain is greater than that of the large slope terrain, and the slope height affects the wake recovery speed.The simulation results of the topographic flow field structure of the tableland with a slope height of 200 and a slope of 30 are shown in Figure 5 under the condition that the inflow wind direction is 45°.Due to the obstructive compression of the mountain, the windward foot of the tableland topography produces a large low-velocity area, and the velocity is lower than that of the leeward return area.
Figure 6-Figure 8 summarize the wind speed distribution at 120 m above the ground on the vertical profile perpendicular to the incoming flow, respectively.Table 3 counts the maximum and minimum values of wind speed at 120 m above the ground in the vertical profile of the terrain at 45 degrees.In the low-height platform terrain, the platform effect is obvious, and the wind speed at the tableland does not change much, and with the increase of slope and height, the wind speed map is similar to the solitary peak terrain.The increase of slope and height will aggravate the blocking effect in front of the mountain and the wake effect in the rear, leading to a reduction in wind speed both ahead of and behind the mountain, and in general, the influence of height on wind speed is greater than that of slope.

Analysis of the Topographic Flow Field of Hogback Ridge
The topography of the hogback ridge ridge is similar to that of the terrace, and the maximum wind speed occurs at the top and two ends of the platform, and the middle of the platform presents a symmetrical parabolic wind speed distribution.Figure 9 shows the distribution of wind resources at a height of 700m and a slope of 20° at a distance of 120m above the ground.Figure 10 and Figure 11 summarize the wind speed distribution at 120 m above the ground in the vertical profile at a 20° slope.Table 4 summarizes the maximum and minimum values of upwind and downwind wind wind speed and their occurrence locations under various working conditions of hogback ridge topography.
At the same height (400m), the smaller the slope, the slower the recovery of the wake behind the mountain, because the smaller the slope, the longer the length of the mountain.When the slope is the same (20), the higher the height, the more obvious the acceleration effect at the mountain top, but the wake effect behind the mountain is also more obvious, and the recovery level of wake is low.Overall, height is more impactful.In this study, the distribution of the flow field of the 0° inflow wind to the terrain of the lower hogback ridge ridge was studied, and the velocity contour and wind speed map are as follows.The wind acceleration effect at the top of the hogback ridge is stronger than that at 270°, because it is more strongly squeezed by the mountain.Since the length of the short axis of the hogback ridge is much smaller than that of the long axis, the downward velocity trend of the 0° wind is similar to that of the solitary peak topography.Figure 12 shows the distribution of wind resources at a height of 700m and a slope of 20° in 0° incoming wind direction at a distance of 120m above the ground.Figure 13-16 summarize the wind speed distribution at 120 m above the ground in the vertical profile in 0° incoming wind direction.Table 5 summarizes the maximum and minimum values of upwind and downwind wind wind speed and their occurrence locations under various working conditions of hogback ridge topography.

Results and Discussion
This paper conducts numerical simulations of the flow field over platform topography and hogback ridge topography.The corresponding evolution patterns of the flow field are as follows: (1)Under the same inflow direction, the distribution of wind resources at each point of the same size terrain satisfies the Reynolds number irrelevance principle, that is, the wind acceleration factor remains unchanged under different inflow wind speeds.
(2) In the low-height platform topography, the wind speed at the platform does not change much, and with the increase of slope and height, the wind speed spectrum is similar to the solitary peak terrain.Under the same inflow condition, taking the slope height of 50m as an example, the downwind wind speed of the small slope terrain is greater than that of the large slope terrain, and the slope affects the wake recovery speed.When the inflow direction is 45°, the low velocity area of the inflow diagonal is obvious compared to the other diagonal.
(3) In the hogback ridge ridge topography, when the height is the same (400m), the smoother the slope, the slower the recovery of the wake behind the mountain, because the smaller the slope, the longer the length of the mountain; When the slope is the same (20°), the higher the height, the more obvious the acceleration effect at the mountain top, but the wake effect behind the mountain is also more obvious, and the wake recovery level is low.The wind acceleration effect at the top of the hogback ridge when the wind direction is 0°, is stronger than that at 270°, because it is more strongly squeezed by the mountain.
(4) Among the topographic dimensions studied in this paper, the distribution of wind resources in the platform topography and hogback ridge topography is more affected by height.

Conclusion
Based on the open-source software OpenFOAM, this paper takes complex terrain modeling and aerodynamic field numerical calculation as the core, and deeply studies the influencing factors of wind resource distribution in two typical complex terrains, and studies the related problems of their variation laws.
This study reveals the influence of incoming wind speed and incoming wind direction on the evolution of the flow field over complex terrain.It can provide references for micrositing in complex terrain and wind power prediction.However, the terrain studied in this research is limited, and the considered influencing factors are also constrained.In the future, coupling with surface roughness, atmospheric stability, etc., can be considered to investigate the evolution patterns of flow fields over complex terrains in diverse atmospheric environments.
a) Calculation of turbulent viscosity; (b) Turbulent Prantl constants controlling k and  turbulent diffusion; (c) Generation and dissipation terms in the k- equation.
k P is the turbulent kinetic energy generation rate due to the time-averaged velocity gradient, b G is the turbulent kinetic energy generation rate due to the buoyancy force, and M Y denotes the dissipation 2023 International Conference on Applied Mathematics and Digital Simulation Journal of Physics: Conference Series 2747 (2024) 012047 IOP Publishing doi:10.1088/1742-6596/2747/1/0120474 of overall turbulent kinetic energy by volume change in the compressible turbulence.k S and S  are defined source terms.

Figure 2 .
Figure 2. Schematic of Hogback Ridge Terrain H400 Slope 20° Terrain Modeling.The dimensions of the mountains studied are shown in Table1

Figure 3 .
Figure 3. Distribution of wind resources under a slope of 30° and an incoming wind speed of 4.5 m/s in the tableland topography of H200.

Figure 4 .
Figure 4. Wind speed variation spectrum at 120 m above the ground in the vertical profile of y=0 tableland topography.

Figure 6 .
Figure 6.Summary of wind speed spectra at 120 m above ground with a slope of H50 and a vertical profile of 10° of tableland topography.

Figure 7 .
Figure 7. Summary of wind speed spectrum of tableland topography with a slope of H200 at a slope of 30° and a vertical profile of 120 m above the ground.

Figure 9 .
Figure 9. Distribution of wind resources at 20° incoming wind speed of 4.5 m/s in H700 terrain of hogback ridge ridge terrain.

Figure 10 .
Figure 10.Summary of wind speed spectrum of topographic height and inflow wind speed changes at hogback ridge ridge.

Figure 11 .
Figure 11.Summary of wind speed atlas of the topographic slope and inflow wind speed change of hogback ridge ridge.

Table 4 .
Summary of wind speed changes in the upwind and downwind directions of the hogback ridge topography.

Figure 12 .
Figure 12.Distribution of wind resources in the slope of H700 at 20° and at 0° and wind speed of 4.5 m/s.

Figure 13 .
Figure 13.Summary of wind spectrum of H200 slope of hogback ridge ridge terrain at 20° and different wind speeds in 0° incoming wind direction.

Figure 14 .
Figure 14.Summary of wind maps of H400 terrain with a slope of 20° and a wind speed of 0° incoming wind direction.

Figure 15 .
Figure 15.Summary of wind spectrum of H400 terrain with a slope of 30° and a wind direction of 0° at different wind speeds.

Figure 16 .
Figure 16.Summary of wind spectrum of H700 with a slope of 20° and a wind direction of 0° at different wind speeds in the back ridge terrain of hogback ridge.

Table 1 .
Typical complex terrain size table.

Table 2 .
Summary of wind speed variations up and downwind of the terrace topography.

Table 5 .
Summary of wind speed changes in the upwind and downwind directions of the hogback ridge ridge terrain.