2.1. Governing Equations of Flow
Fluid dynamics is controlled by three conservation laws of classical mechanics, namely, the conservation of mass, momentum and energy. Partial differential equations (PDEs) can be derived from these conservation laws and simplified in appropriate cases. These conservation laws are derived under the assumption of continuum mechanics, and we can then express the physical characteristics of flow (including velocity, pressure, density, etc.) as time-dependent scalars or vectors in three-dimensional space
, such as pressure
and velocity
. For liquids, such as the water used in this article, the continuity assumption always holds in practice.Flow can be represented in the Eulerian formulation as a time history of flow properties at each fixed point in a space domain. According to three laws of conservation, the continuity, momentum and energy equations in an arbitrary material volume
with surface
are derived [
13], they (totally called general N-S equations) are as follows, herein Einstein summation convention (1916) is used.
where,
is the stress tensor,
is the body force per mass acting on a material particle,
is the total energy per unit mass,
e is the internal energy per mass,
is the kinetic energy per mass of a material particle,
q is the rate of heat added to each material particle per mass. If
s is the heat flux per area through the surface
, heat diffusion is governed by Fourier’s law, i.e.,
grad
T, wherein
k is the thermal conductivity and
T is the temperature. Assuming Newtonian fluid, we can get the simplest constitutive relation, that is, the relationship between stress tensor and fluid motion [
14]
where,
,
,
p is the pressure,
is the Kronecker delta,
is the dynamic molecular-viscosity coefficient,
is the rate of strain tensor. When dealing with continuous medium Newtonian fluid, the influence of molecular fluctuation can be smoothed out through viscosity and represented by
. To take assumptions that flow is incompressible (Mach number
,
1.4 km/s in water at 15 °C, depending on the amount of dissolved air), and the density of the water is constant, i.e.,
(
kg/m
3 4 °C). Equations (1)–(3) can be simplified as
Equations (6) was first derived by Navier (1823, Frenchman), Poisson (1831), de Saint-Venant (1843) and Stokes (1845, Englishman). Since meteorologist Richardson, numerical schemes have allowed us to solve motion equations in a deterministic manner after providing initial conditions and specifying boundary conditions [
15].
When deriving the general N-S equations, it was not assumed that the flow was laminar or turbulent, but laminar flow was stable, and turbulence was chaotic and diffusive, leading to rapid mixing, time-dependent, and three-dimensional (3D) vorticity perturbations with a broad scale range of time and length [
16]. The most accepted theory about turbulence is based on the energy cascade concept developed by Kolmogorov [
17,
18], according to which turbulence is composed of eddies of different scales. Large eddies break up and transfer their energy to small eddies in a chain process. At the smallest eddy scale, molecular viscosity can effectively dissipate turbulent kinetic energy and convert it into heat. The smallest eddy is characterized by the Kolmogorov micro length scale
and time scale
with
the molecular kinematic viscosity and
the average rate of dissipation of turbulent kinetic energy (TKE). The largest eddy is proportional to the size of the geometry involved, also known as the integral length scale.
Turbulence DNS simulation based on the energy cascade concept must use a very small time step to solve the whole spectrum of turbulence scales contained in time and space. This time step is determined by the Courant number , to ensure that , resulting in a very large number of grid points (proportional to Reynolds number, , ). Due to the extremely high and unacceptable computational cost of DNS simulation, it is currently not widely used to solve industrial problems, and this situation may also change in the future with the development of computer technology.
To reduce significant computational costs, statistical analysis is used to simplify the resolution of turbulence. The time-dependent nature of turbulence and its wide range of time scales indicate that statistical averaging techniques can be used to approximate random perturbations. At present, the most popular method for solving turbulence in industry is RANS simulation [
19], where statistical averaging is based on an appropriate time, and the requirement for the number of grid points is not as high as DNS and LES simulation. With the development of computer technology in recent years, the use of LES to solve industrial problems is becoming increasingly acceptance. The statistical average of LES simulation is different from that of RANS simulation based on spatial average [
20]. There are three Reynolds averaging techniques [
19], i.e., time averaging (TA), spatial averaging (SA) and ensemble averaging (EA).They are shown as followings respectively
where,
N is the number of individual experiments. TA is the average of physical quantities over a time period and is suitable for steady-state turbulence. The average value of steady-state turbulence over time period
T does not change over time. SA represents the average of physical quantities on a spatial volume
V, suitable for homogeneous turbulence. EA is the overall average over a certain time, applicable to any type of turbulence. EA is the average of many identical quantities at a certain time and is suitable for any type of turbulent flows. If
T is sufficiently large compared to the turbulent perturbation time scale, the Equation (
8) can be replaced by
LES simulation began with the famous Smagorinsky model model in the 1960s. Meteorologist Smagorinsky wanted to express the influence of three-dimensional subgrid turbulence cascading to small scales mechanism on large-scale synoptic quasi two-dimensional atmospheric or ocean movement. This mechanism was described by Richardson in 1926 and formally proposed by the famous mathematician Kolmogorov in 1941. In the 1970s, theoretical physicist Kraichnan developed an important concept of spectral eddy viscosity. Since then, LES simulation research was first carried out in schools Stanford-Torino and Grenoble, and then many researchers from around the world studied it. The calculation of the first LES simulation dates back to the 1960s, but the rigorous derivation of the governing equations for LES simulation in a general coordinate system was published in 1995. There are various ways of modeling the subgrid terms, and a lot of models existing. Sagaut [
20], a passionate researcher in LES research, clearly defines the various models currently used as two categories: functional model and structural model.
After determining the approximation level, we can define the mathematical model of the physical system to be described. The so-called approximation level is to obtain the required accuracy on a fixed parameter system. In a macroscopic description, the corresponding scale is much larger than the mean free path of molecules, and the continuum paradigm is invoked. In this paper, we study the single-phase (water) incompressible viscous Newtonian fluid, assuming that the density of water is constant, without considering external body force, and without solving the energy equation. Under the framework of continuum mechanics, the most complete model is the general N-S equations, which uses state equations and empirical laws that describe the dissipation coefficient as other variables. Under afore mentioned assumptions, the general unsteady N-S equations can be rewritten in the physical space as
where,
p(=
) and
are the static pressure and the assumed constant kinematic viscosity respectively. They are the continuity and momentum equations with velocity field
expressed in a Cartesian coordinates system
, which is the Eulerian velocity-pressure formulation. In order to achieve a well-posed problem, initial and boundary conditions must be added to the system later.
2.2. LES Models
Based on different levels of spatial-temporal resolution and dynamic description, LES simulation is directly applied to solve low-frequency spatial modes. The constitutive equations of LES simulation, i.e., the filtered N-S equations, are as follows
where,
is the filtered pressure. The filtered momentum equation introduces a nonlinear term
which can be represented by a function of
and
, and then we obtain the nonlinear term:
where,
is the cross-stress tensor representing the interactions between large and small scales, and
is the Reynolds subgrid tensor reflecting the interactions between SGS. The
term can further be decomposed as
. The newly introduced Leonard tensor
represents large-scale relationships. If the filter commutation error under spatial differentiation can be ignored, the filtered momentum equation can be expressed as:
where,
is the subgrid tensor defined as
which is the famous Leonard or triple decomposition.
LES simulation is based on the separation of large and small turbulent scales, and anisotropic large scales containing higher energy can be directly simulated and calculated, while small scales must be modeled using subgrid scale models. Turbulent small-scale fluctuations are modeled through eddy viscosity and diffusivity assumption. The scale greater than the cut-off length of the filter is called large or resolved scale, while others are called small or subgrid scale (SGS). The SGS model is a statistical subgrid model, and precise modeling of SGS is possible because small scales smaller than the filter cutoff length can be considered isotropic and do not depend on the type of flow and boundary conditions. Large eddy simulation is the decomposition of the nonlinear term in the above equation, and then modeling the unknown term. Various modeling strategies and models already exist, and the most popular SGS model is the algebraic eddy viscosity model, proposed by Smagorinsky [
21], i.e.,
where,
is the SGS eddy viscosity,
is the Smagorinsky constant,
is the resolved strain rate tensor with
. The constant
must be adjusted for different flows. This inconvenience can be skipped by using a dynamic procedure proposed by Germano and et al. [
5], in which two filters are used to directly solve the model coefficient
, namely a grid-scale filter
and a test filter
(typically,
). By applying the test filter, we can obtain
Assuming that the subgrid stress has self-similarity, we can model
in the same way as modeling
, that is
with
, and to combine Equations (19)–(22), we can get an equation for
as
where,
. Equation (
23) for solving one unknown
is an overdetermined equation system. To minimize the error, Lilly [
6] proposed a least squares method to solve it, which can obtain
The dynamic modelling combined with Smagorinsky model is named dynamic algebraic SGS model (LES-DASGS), which can also be called dynamic Smagorinsky model (LES-DSM). The calculation of coefficient
can lead to numerical instability, as the value of the denominator in Equation (
24) may become very small at certain points in the flow, and there may be persistent negative values of
at some locations. This problem can be solved by taking the local mean of the model coefficients along the direction of homogeneous flow [
5]. Meneveau and et al. [
22] proposed an average technique along particle trajectories. Kim and Menon [
23] developed another local averaging technique based on the local structure of vorticity. Whether based on direction of homogeneity or local structures, the local averaging technique is an artifact (without any physical justification) to avoid numerical instability and isn’t consistent with the dynamic procedure. A true dynamic model should solve the model coefficients locally without using any averaging techniques. However, research by Ghosal and et al. [
24] has shown that when solving flows with homogeneous directions, the results obtained using averaging techniques are the same as those obtained using more rigorous dynamic models, but the computational cost is lower. Ghosal and et al. [
24] also proposed another local dynamic model for inhomogeneous flows.
Applying a dynamic
k-equation subgrid-scale model belonging to subgrid viscosity models, we can obtain the LES-DKSGS model, that is [
25,
26,
27]
where,
is the SGS kinetic energy. The three terms to the right of Equation (
25) are the production rate, the dissipation rate and the transport rate of
respectively. SGS tensor
can be modeled as
where, if the cut-off length of the filter is within the applicable range of the scale similarity assumption, there are
and
. The establishment of this model does not assume a local balance between SGS energy generation and dissipation, but instead directly calculates SGS. Therefore, the model can consider non-local and temporal effects, which are ignored in LES-DASGS model. It is optimistic to expect LES-DKSGS model to provide better calculation results than LES-DASGS model. Using the same procedure, the equation for
can be derived as
with
and
. The TKE
K at the test filter level is obtained from trace of Equation (
21). Equation (
27) has the same form as the Equation (
23), so
is also determined using the least-square method, that is
The following equation can be obtained based on the relationship between the dissipation rate between the grid scale filter (
) and the test filter (
)
with
and
. Then, we can further obtain
with
. The Equation (
30) is a scalar equation for one unknown
and can be solved directly as
. The model coefficients in this LES-DKSGS model is also evaluated by using a local averaging technique to avoid numerical instability and share the same defects as LES-DASGS. To address this issue, Kim and Menon [
23] proposed a local dynamic
k-equation subgrid-scale (LES-LDKSGS) model where the model coefficients are calculated locally without using any averaging techniques or causing any numerical instability. As shown in
Table 1, we consider scales with different energy levels. The filtered kinetic energy characteristic length scale
is unknown and is related to the dissipation rate, and there is a relationship
due to relationship
. Term
contains SGS kinetic energy, while
is related to the rate of SGS kinetic energy generation, and there is a clear separation between the scales of SGS kinetic energy generation and dissipation. To accurately model the generation and dissipation of kinetic energy in SGS, additional information on energy transfer at these two length scales must be provided, but this information is unknown. Therefore, we further assume that the energy transfer that occurs on a small scale is determined by the strain rate on a large scale and requires bidirectional energy transfer. Therefore, we use length scale and large-scale strain rate to model the generation and dissipation of SGS kinetic energy.At the testing filter level, we can conduct similar derivation and discussion to obtain
. Considering three different SGS stress tensors and dissipation rates at one grid filter level and two other test filter levels, it can be obtained that
with
and
where, the denominator of Equations (37) or (38) always contains energy information of resolved scale that will not be zero. This dynamic model is valid within the applicable range of the scale similarity assumption. The scale similarity between SGS stress
and resolved stress
is supported by Liu and et al. [
28] research results at a reasonable high
. The proposed LES-LDKSGS model is more effective, cheaper and robust than LES-DASGS.
2.3. Geometry and Performance Characteristics of Propeller
This study used a five bladed, right-handed, variable pitch, non skew, and no rake marine propeller DTMB 4381, as used in Bridges [
29]. Cylindrical cross-sections of the blade at
are airfoils as shown in
Figure 1, in which
R is the radius of the propeller rotor disc. The distributions of relative chord length (normalized with propeller diameter
D) and twist angle are shown in
Figure 2. The hub was tapered.
The forces and moments of a specific geometric propeller can be represented by a series of dimensionless quantities [
30], including thrust coefficient
, torque coefficient
, advance coefficient (also called advance ratio)
J and cavitation number
, they are
where,
e is water vapor saturation pressure. The dimensional analysis method can be used to analyze geometrically similar propellers. When a marine propeller operates far from the free water surface and does not cause surface waves, the forces acting on it are easily understood to be related to the diameter
D, the advance speed
, the rotational speed
n, the density of the fluid
, the viscosity of the fluid
and the static pressure of the fluid at the propeller
. Therefore, the thrust
T on the propeller is proportional to them, i.e.,
. To ensure that the dimensions on both sides of the equation are the same, we can obtain
, with the mass
M, the length
L and the time
T. Compare the left and right ends of this equation to get
for
M,
for
L and
for
T, then we get
and
, which used to get
, then
, so
In the same way we also have
. The open water efficiency of a propeller is defined as the ratio of thrust horsepower to delivered horse power as
Let
the brake power or the maximum continuous rating power which is delivered at the engine coupling flywheel, then the shaft power
at the output coupling of the gearbox is
with
. The power transmitted to the propeller and absorbed by the propeller is defined as the increase in kinetic energy per unit time of the slipstream or the work done by the propeller thrust after deducting the loss of bearings, i.e.,
with
, in which the shafting mechanical efficiency
is usually between 98% and 99% depending on the length of the shaft and the number of bearings. Another definition of thrust and power coefficient is
with the disc area
. International Towing Tank Conference (ITTC,1978) report another Reynolds number
with
the chord length at
of the propeller.
2.4. Computational Domain and Meshing
To avoid blocking effects [
31] caused by setting the computational domain too small, set the right-hand reference coordinate system and arrange the computational domain as shown in
Figure 3. The outer boundary of the flow field, namely the water channel, is a cylinder with a diameter of
and a height of
. The center of propeller blades is at the origin and the flow direction is in the positive direction of the
x-axis. The computational domain spans between
upstream and
downstream of the propeller. The channel blocking factor for this arrangement is about 0.021 (less than 0.1), where the blocking factor of the channel is defined as
with
A the disc area of propeller and
the area of water tunnel cross-section. The simulations are performed using unstructured grids with about 8.5 million hexahedral cells for LES-LDKSGS and RANS-kEpsilon (also with different refinements). Views of overall mesh distribution are shown in
Figure 4. Grid refinement is performed in the wake zone of the propeller to capture the small-scale structures of interest to us. The grid scale size in different regions in
Figure 3 is shown in
Table 2. The time step for LES numerical simulation is set to
s, and the time taken for the propeller to rotate one cycle at the designed advance ratio
is equivalent to 3956 calculation time steps, and at
J = 313,350 calculation time steps.
Considering the computational cost, LES-LDKSGS simulation was only performed on approximately 8.5 million grids with
and
. In order to compare and illustrate, we also intentionally performed simulations on about 8.5 million, 3.07 million, 1.15 million and 38,000 grids for RANS-kEpsilon at different advance ratios, although there is no need for such fine mesh (such as 8.5 million) in RANS-kEpsilon simulation with a good grid strategy. All 42 simulations included in this paper are listed in
Table 3.
2.6. Numerical Method
Use unstructured mesh finite volume method for numerical calculations, and use numerical altorithms to solve partial differential equations (PDE). The semi-implicit method for pressure-linked equations method (SIMPLE) [
32,
33] widely known in the engineering literature is perhaps the oldest and most widely used iterative method for the stationary N-S equations. For unsteady N-S equations we use pressure implicit with splitting of operators (PISO) method [
34], which is a variant of SIMPLE. In PISO method which is a predictor-corrector approach, we take the small-time-step assumption so that the pressure and velocity coupling is much stronger than the nonlinear coupling. Then the nonlinear term can be linearized and the velocity field can be explicitly corrected, while the discretization of the momentum equation is safely frozen. The solver is set up on the platform of OpenFOAM [
35] and mainly includes two steps: the momentum prediction and the pressure correction.
Firstly, calculate the velocity at the center of the cell, and then interpolate its value to the center of the cell surface, which is called the flux. Then project the flux to ensure that the pressure Poisson discrete equation is satisfied. Using a multigrid approach to iteratively solve the Poisson equation, the obtained pressure value is used to update the predicted velocity field. Time discretization using a second order implicit Crank-Nicolson bounded scheme. The standard finite volume discretization of Gaussian integral is used for spatial discretization, and linear interpolation is used for the value from the center of the cell to the center of the face. And limiting the gradient so that when the calculated gradient is used to extrapolate the cell value to the face, the face value will not exceed the range of values in the surrounding cells, thus ensuring boundedness. In order to maintain the second-order accuracy, explicit non orthogonal correction is taken for non orthogonality grids. The pressure Poisson equation is solved using the geometric aggregated algebraic multi-grid (GAAM) method.