Hydrodynamics is the branch of fluid mechanics that studies the motion of isovolume fluids and the forces acting on solid bodies inmersed in it. These analysis is performed in order to find the solution of problems like the formation of freak waves and its impact with structures. Usually, the definition of these problems implies a set of differential equations hard to solve analytically, therefore its solution and simulation is seek using experimental or nummerical ways.
Viscous flow approach:
A fluid is a continuous medium where physical quantities can be defined locally, in which the quantities of interest (temperature, pressure, velocity, density…) are supposed to vary continuously.
The set of conservation equations that gather the variation in these quantities in order to describe the behaviour of a Newtonian fluid, that is to say a fluid with viscosity, are the Navier-Stokes equations.
- Equation of state: local thermodynamic equilibrium of the fluid.
- Continuity equation: conservation of mass.
- Conservation of momentum: fundamental law of dynamics.
- Energy balance equation: conservation of energy.
A fluid is assumed to be isovolume (incompressible) when its density is not dependant from pressure or temperature. After this assumption the Navier-Stokes equations can be simplified to a large extent.
Boundary conditions:
The system of equations used to analyse the fluid behaviour inside a domain are the Navier Stokes adding boundary conditions and initial conditions.
- Kinematic Free Surface Boundary Condition: assumes a continuity of the free surface (no wave breaking).
- Dynamic Free Surface Boundary Conditions: is based on force balance at interface. States a dynamic pressure distribution at the 3 domain axis. In Ocean Enginnering the influence of surface tension within this boundary is negligible since the size of the structures are large enough.
- No slip condition: for a viscous Newtonian fluid, assuming that the velocity at the sea bottom is 0 and increments along its velocity profile.
- Symmetry condition and others.
Potential flow approach:
An inviscid fluid is the one which has no viscosity. This means that the boundary layer effect is neglected and the fluid at a solid boundary has a free slip condition, which entails a discontinuity in the velocity profile near the body surface.
The lack of friction between the flow and the solid boundary implies that the generation of vorticity, which is a local rotation of the fluid particles, is not produced. So the particle vorticity will be 0 or constant.
Euler model: is applied for high Reynolds flow around a body where boundary layer and wake are thin, hence the influence of viscosity can be neglected. Despite regarding the fluid as inviscid, the Euler equations contemplate a value of vorticity different of 0. So, the flow in Euler model is assumed to be inviscid, incompressible and rotational.
Potential theory: now apart from inviscid and isovolume, the flow becomes irrotational.
Viscous flow approach VS Potential flow approach:
In this section, the usuall hydrodynamic problems simulated with CFD software are going to be analyzed, and in this way define the most appropiate approach for its simplification.
Wave propagation
Is a linearized regular wave model propagating in an infinite water depth without the interaction of a solid wall. It is caraterized by a Reynolds number range high enough to entail a lack of wave breaking. In addition, there is no solid interference of a structure o sea bottom. Therefore, the application of a potential flow approach can be fullfilled, since viscous effects are negligible and any vorticity is generated.
Ship resistance:
Ship with forward speed in initial calm water. The flow is characterized by a high Reynolds number that indunces large turbulent effect. Turbulent flows reduce the generation of flow dettachement, which is the main factor that involve the use of viscosity approach. Therefore, viscosity can also be neglected, but in this case wave breaking can occur at the ships bow and vorticity can be found. Resuming, potential and viscous approaches are acceptable for this problem, but at the time of evaluating hydrodynamic forces like drag, viscosity can not be neglected and viscous approach is needed.
Seakeeping:
Is the response of a marine structureagainst the influence of waves. It simulation is not very common to see within the industrial sector. The applied approach depends on the size of th structure, which is determined by the Keulegan-Carpenter number.
- Large bodies (Kc < 2): viscous effects are only significant near the body, since there is any flow dettachement, in general there is a low influence of viscosity and vorticity, hence potential approach can be used.
- Small bodies (Kc>10): contrary to large bodies, in small bodies large flow dettachement is generated leading to locally large viscous effects and vorticity. Thus, viscous approach needs to be applied.
Maneuverability:
When a ship moves or changes it direction in calm water, wave breaking can occur in the vicinity of its hull. Also large flow dettachment due to small components like rudder or keels is possible, this is why viscosity and vorticity have a big influence, above all in the ship wake and appendages. Viscous approach is the most appropiate for its simulation, nevertheless it is not widely used in the industry.
Impacts and wave breaking:
These phenomenons are likely to take place in ship slamming or waves interacting with structures. The flow generated is characterized by having a complex shape including vortices, bubbles or drops. Consequently, the viscosity will have a big influence at the time of simulating the shape of the flow and a viscous approach will be needed.
Lifting profiles and propellers:
The analysis of a propeller or a profile is performed without the boundary contition of free surface. Since turbulent and non turbulent flow can appear, the approach used will depend on the flow angle of attack over the profile.
- Potential flow can be modeled only for the case of low angle of attack since there is not a big detachment of the boundary layer, so the effect of viscosity will not have a significant influence. For that reason, the flow can be simulated using potential theory.
- In the case of high angle of attack, the presence of detachment in the boundary layer at the beginning of the profile entails an influence of viscosity which is also translated in a presence of rotationality, therefore viscous flow theory needs to be applied.
Nummerical methods:
Analytical solution is difficult to get when geometries are complicated or the physical phenomenon is complex or non-linear, as is the case of a fluid behaviour. That’s why we need a numerical method to be able to solve a set of partial differential equations representing the evolution of a physical system.
Mesh discretization:
The principle of a mesh is to discretize the fluid domain solution of the fluid equations at a finite number of fluid points. There are different mesh types:
- Structure Mesh: easy and more accurate. It is possible to use cells with higher slender ratios, controlling the shape and densifying the mesh where the variations are more important, like in the boundary layer.
- No-Structure Mesh: provide lower accuracy that the structured but represents better the body.
- Hibrid Mesh: is mix between structure and unstructured mesh.
For a calculation around an usual ship (no appendages, no propeller) around 5 million nodes are required for the grid.
Hydrodynamic mathematical models:
- RANS (Navier-Stokes)
- Euler equations
- Potential flow
Set of equations to define te problem: RANS equations (modified transport equations) + Mass continuity + Free Surface Boundary Conditions (if mono-valuated) + Other boundary conditions + Initial conditions
Main applied N-S methods in hydrodynamics:
- Boundary element Method: wave/large bodies interaction with no forward speed, using to calculate hydrodinamic coefficients acting on a body for seakeeping. The general principle of this method consist in the panel discretization of the domain boundaries (free surface, immersed part of the hull, lateral walls…). In each panel a distribution of singularities is located, each singularity generates an elementary flow, by the combination of elementary flows a realistic one can be computed. The results given are the wavefield around the hull and its deformation and the wave coefficient.
- Mesh-based CFD (Euler or RANS): mostly used to calculate ship resistance in calm water simulating the viscous flow around the ship, with not too complex free-surface. They are an approximated simplification of the Navier Stoques equations in order to provide a turbulence modeling. It is performed considering the velocity and pressure of the flow are determined as the sum of a mean (-) and a fluctuating quantity (~). Then, these approximations are introduced in the instantaneous Navier-Stokes equations. Later, assuming that the average of the fluctuating quantities is zero, the equation can be simplified. Finally, the RANS equations obtained are similar to the N-S but the quantities involved are mean quantities and have an additional Reynold’s stress term.
- Spectral method: is used to simulate the wave evolution by means of a potential approach.
- SPH: complex free-surface flows, fast dynamics flows.
Verification and validation:
Verification: consists of the following conditions.
- Consistency: it has to compute the same solution for the same initial conditions and input parameters, if not it is chaotic.
- Stability: the results can not tend to infinity.
- Convergence (in time and space): refers to the scheme providing a result that is close to the analytical solution when Δt→0 and Δx→0. That means ideally with an infinitesimality small time step and very fine mesh, the results should converge to the analytical solution.
Validation: is subjective, it can be assessment through comparison using external references. A small error percentage can be very bad for some cases and fine for other ones, the same happens for high percentage errors.
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