Application for unsteady flow
and the drag effects. Taking a dam-break problem, whose unsteadiness is supposed to be similar to the actual phenomena such as tsunami, as an example problem, we carry out the macroscopic flow simulations with the 2D shallow water equation incorpo-rated with either the roughness coefficients (Chapter 3) or the surrogate model of the momentum loss parameter (Chapter 4). Comparisons between the analysis results with the 3D direct numerical simulations offer the discussion for the capabilities of the proposed multiscale evaluation schemes and afford an insight into the significance of the inertial effects.
5.2 Governing equation for 2D macro-scale unsteady flow
According to Suzuki et al. [2019], the macro-scale shallow water equation for unsteady flow can be defined by implementing the inertial term into the left hand side of the momentum equation (2.1) as:
ρ (
(1 +Cm·m)∂Ui
∂t +Uj∂Ui
∂x˜j
)
=−ρg∂h
∂x˜i −Si
∂h
∂t + ∂
∂x˜i
Uih= 0
in Ω , (5.1)
where m means the volume fraction defined by following formula:
m= Ytree
Y , (5.2)
whereYtree,Y represents the domain of water and trees as defined in (2.4) at previous section. Here, Ytree is a variable that depends on the submerged depth. This volume fraction m has been generally substituted by the volume of cylinders per unit area because of avoiding the cumbersome measuring procedure. We calculate m from the following formulation:
m=YtreesingleN, (5.3)
where N represents the number of trees per unit area, Ytree means the volume of a single rigid tree. Fig. 5.1 shows quantity of Ytreesingle at each submerged depth. The volume Ytree is precisely measured from a pvpython code with FE mesh information of trees. The Cm in the left-hand side of (5.1) is inertial coefficient. The evaluation of this inertial coefficientsCm is provided from the next section. Here,Si is a source term that represents the presence of trees. The definition of Si is based on either the energy balance-based (Chapter 3) or the momentum balance-based multiscale modeling (Chapter 4) as:
Si =
TB,i
h : Energy balance-based modeling 1
2ρC|U|Ui : Momentum balance-based modeling
. (5.4)
With the appropriate modeling of source term Si, (5.1) allow us to simulate the macro-scale unsteady flow such as tsunamis more precisely.
0.22 0.20
0.15
0.10
0.05
0.00
h [m]
0.0 5.0e-6 1.0e-5 1.5e-5 2.0e-5 2.0e-5 Volume of a single rigid tree model
Figure 5.1: Volume of a single tree depend on submerged depth h
5.3 Evaluation of inertial coefficients
To calculate the inertial coefficient Cm which is the essential parameters to simulate the unsteady state-flows by the governing equation (5.1), we utilize the results of numerical flow tests provided in previous chapter Chapter 4.
Although we defined the equilibrium between the macroscopic pressure gradient Px1 and the total momentum loss inside the LTD in Chapter 4, we confirm that there are unmissable gaps between thePx1 values and the momentum loss values reproduced by the surrogate model in Fig. 4.10. Though a part of them seems to be caused by interpolation errors, we reasoned that most of this difference arises from the absence of the inertial effects. So we evaluate the inertial effects from the following residual calculated from the numerical flow tests results:
Finertial =Px1 − 1
2ρC⟨u⟩⟨u1⟩, (5.5)
where is equal to the difference between the two values plotted in the graphs provided at Fig. 4.10.
Considering that the classic formulation for the fluid force acting on the submerged structures (Morison et al. [1950]), the velocity acceleration seems to dominate the amount of inertial forceFinertial. So we calculate the velocity acceleration a posterior to identify the relationship with the inertial effects. Here, the acceleration of global velocity is derived from the following equation:
∆⟨u1⟩
∆t
t
= ⟨u1⟩t+∆t− ⟨u1⟩t
∆t , (5.6)
where ⟨u1⟩ means the streamwize component of volume averaged velocity measured and are provided in Fig. 4.6. By setting ∆t to be 0.1 [s], time histories of ⟨u1⟩
∆t
t
are obtained as shown in the black solid lines at Fig. 5.2. As well as the acceleration
⟨u1⟩
∆t
t
, the inertial effects calculated from the (5.5) are plotted in the dashed-line.
A comparison between solid lines and dashed lines indicates the correlation between
Finertial and ∆⟨u1⟩
∆t . Both values tend to increase at the beginning of the flow simula-tions dramatically. After they reach the peak values, the variation of both amounts becomes calm, and the flow seems to be steady-state. These similarity confirm us the correlation between the inertial effects Finertial and the velocity acceleration ∆⟨u1⟩
∆t . Further observations to the graph provided in Fig. 5.2 reveal that ∆⟨u1⟩
∆t are almost under 0.1 [m/s2] when the flow seems to be steady state. From that, we consider that the Finertial recorded when the ∆⟨u1⟩
∆t is over 0.1 should be dealt with as the data of inertial effect-dominant. We extract those inertial effect-dominant data and summarize in Fig. 5.3. The horizontal axis, vertical axis means the Finertial and
∆⟨u1⟩
∆t , respectively. We can observe the linear correlation from the graph at Fig. 5.3 although the data has dispersion. The colors of each plot mean the Froude number Fr that calculated from the following equation.
Fr= √⟨u1⟩
gh (5.7)
From the observations to Fig. 5.3, Fr seems to be not so influential to Finertial. We then conclude thatFinertial can be summarized as an approximation function obtained from the multiply of ∆⟨u1⟩
∆t and the inertial coefficient as:
Finertial ≈ρCm∆⟨u1⟩
∆t [N/m3]. (5.8)
The black-solid lines in Fig. 5.3, which is a linear function that identified by the least squares method, can reasonably explain the tendency of the relationship between
∆⟨u1⟩
∆t and Finertial. The coefficient of determination R2 is 0.6261 and is relatively close to 1.0. We obtain the inertial coefficients Cm=0.42550 as the gradient of the linear function. Comparing with the values that is generally used in the conven-tional (Sumer et al. [2006], Suzuki et al. [2019]), this value is relatively small. It can be conclude that the previous simulations overestimated the inertial effects on the submerged structures.
In the next section, we employ this evaluated inertial coefficients in 2D shallow-water flow simulations (5.1) and then investigate the importance of the inertial effects.