## 2 Materials and Methods
### Numerical Calculation  
In order to understand the turbulent statistical characteristics of the vertical jet energy dissipator, the three-dimensional (3D) height and turbulent statistical characteristics of the vertical jet were simulated by numerical simulation. The effectiveness and accuracy of the wall-modeled large eddy simulation (WMLES) in simulating the height of the jet were verified by the experimental results. The large eddy simulation belongs to the fully developed numerical turbulence calculation method with a high Reynolds number [40]. The jet studied in this paper belongs to a high-speed jet, which is calculated by using a wall-modeled large eddy simulation (WMLES) turbulence model. WMLES is scalable and applicable to a wide range of contexts, particularly for high Reynolds number flows and complex geometries. However, its scalability is limited by computational costs, and its applicability depends on the accuracy of wall models and SGS models in different flow regimes. With ongoing advancements in computational power and modeling techniques, WMLES is likely to become even more scalable and applicable in the future. For specific applications, careful validation and adaptation of the method are essential to ensure reliable results. In the flow direction measured from the nozzle, Z is the vertical coordinate, X is the lateral coordinate, and Y is the radial coordinate. The vertical velocity is W, the lateral velocity is U, and the radial velocity is V. Figure 6 uses the isosurface \(W/W_{j}\) to show the outer boundary of the shear layer in the instantaneous flow field under the conditions of Re = 32,639, the size of the crushing needles m = 0.15, and the number n = 0. In this study, the height of the jet is defined as the distance between the jet from the outlet to the highest point of the outer boundary of the shear layer. Under dimensionless conditions, the spatial scale of the jet is dimensionless by the ratio of the distance in the Z direction, the distance in the X direction, to the diameter D of the nozzle. The whole model is divided by a structured grid, the grid size of the model is 0.17 D, and the crushing needle area is locally encrypted. The minimum grid size of the crushing needles' area is 0.005 D, and the maximum grid size is 0.01 D. In the numerical simulation, the velocity input is used below the vertical jet pipeline, and the atmospheric pressure is set above the vertical jet pipeline. The outlet boundary is the pressure outlet; the wall is a non-slip wall. Figure 5: The schematic diagram of the crushing needles’ arrangement is (**a**) n = 0; (**b**) n = 2; (**c**) n = 3; (**d**) n = 4; (**e**) n = 5; and (**f**) n = 6. The following assumptions were made in this study:  
* The flow is incompressible and steady. * The fluid is Newtonian with constant properties. * The effect of gravity is negligible. * The flow is fully developed at the inlet. The governing equations for the fluid flow are as follows:  
Continuity equation:  
\[\
abla\cdot u=0 \tag{1}\]  
Momentum equation:  
\[\rho\left(\frac{\partial u}{\partial t}+u\cdot\
abla u\right)=-\
abla p+\mu \
abla^{2}u \tag{2}\]  
where \(u\) is the velocity vector, \(p\) is the pressure, \(\rho\) is the fluid density, and \(\mu\) is the dynamic viscosity. The following boundary conditions were applied:  
* Inlet: Uniform velocity profile, \(u=U_{0}\), \(v=0\);
* Outlet: Zero pressure gradient, \(\frac{\partial p}{\partial x}=0\);
* Walls: No-slip condition, \(u=0\), \(v=0\). WMLES Implementation Details:  
Governing Equations: The filtered Navier-Stokes equations were solved using a finite volume method with a second-order accurate spatial discretization scheme. SGS Model: The dynamic Smagorinsky model was used to account for subgrid-scale turbulence, with model coefficients computed dynamically based on the local flow field. Figure 6: The outer boundary of the shear layer. Wall Model: An equilibrium wall model based on the logarithmic law of the wall was employed, with wall shear stress computed using the Spalding formula.