28
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
form the first derivative is always obtained from the upstream, the formula expression
is
dϕ
dx
i
=
ϕ i − ϕ i−1
δx
, u i > 0
(2.26)
dϕ
dx
i
=
ϕ i+1 − ϕ i
δx
, u i < 0
(2.27)
(2) Definition of control volume integral method
The value of variable Φ on the control volume interface is defined as follows (see
Fig. 2.3b):
On interface e: u e > 0, ϕ = ϕ P ; u e < 0, ϕ = ϕ E (3.28a).
On interface w: u w > 0, ϕ = ϕ w ; u w > 0, ϕ = ϕ P (3.28b).
That is to say, the unknown quantities on the interface always take the values at
the upstream node while the central difference takes the arithmetic mean of those
at the upstream and downstream nodes, and this is the basic difference between the
two schemes.
(3) Meaning of false diffusion
The false diffusion is defined as a phenomenon of relatively large numerical calculation error for the reason that the truncation error of the discrete scheme obtained
by the first derivative term of the convection–diffusion equation is smaller than the
second order.
The partial difference of the first derivative term is the cause of the false diffusion.
From the characteristics of the physical process itself, the effect of diffusion always
reduces the change rate of the physical quantity, which in turn makes the whole field
uniform. In a discrete scheme, the presence of false diffusion exacerbates the extent
to which the results of numerical solutions deviate from the true solution. The greater
the degree of false diffusion, the more serious the deviation. Where the first term of
the truncation error of a discrete equation is an even-order spatial derivative, the error
of the numerical calculation result features diffusion property; where the first term
of the truncation error is an odd-order spatial derivative, the error features dispersion
property.
(4) Second-order upwind scheme
The second-order upwind scheme can be employed to overcome the shortcomings
of the first-order upwind scheme while taking advantage of the benefit of the upwind
scheme.
For the evenly divided grid shown in Fig. 2.4, the second-order upwind scheme
is defined as
u
∂ϕ
∂ x
≈
u i
2x
(3ϕ i − 4ϕ i−1 + ϕ i−2 ), u i > 0
(2.29a)
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
form the first derivative is always obtained from the upstream, the formula expression
is
dϕ
dx
i
=
ϕ i − ϕ i−1
δx
, u i > 0
(2.26)
dϕ
dx
i
=
ϕ i+1 − ϕ i
δx
, u i < 0
(2.27)
(2) Definition of control volume integral method
The value of variable Φ on the control volume interface is defined as follows (see
Fig. 2.3b):
On interface e: u e > 0, ϕ = ϕ P ; u e < 0, ϕ = ϕ E (3.28a).
On interface w: u w > 0, ϕ = ϕ w ; u w > 0, ϕ = ϕ P (3.28b).
That is to say, the unknown quantities on the interface always take the values at
the upstream node while the central difference takes the arithmetic mean of those
at the upstream and downstream nodes, and this is the basic difference between the
two schemes.
(3) Meaning of false diffusion
The false diffusion is defined as a phenomenon of relatively large numerical calculation error for the reason that the truncation error of the discrete scheme obtained
by the first derivative term of the convection–diffusion equation is smaller than the
second order.
The partial difference of the first derivative term is the cause of the false diffusion.
From the characteristics of the physical process itself, the effect of diffusion always
reduces the change rate of the physical quantity, which in turn makes the whole field
uniform. In a discrete scheme, the presence of false diffusion exacerbates the extent
to which the results of numerical solutions deviate from the true solution. The greater
the degree of false diffusion, the more serious the deviation. Where the first term of
the truncation error of a discrete equation is an even-order spatial derivative, the error
of the numerical calculation result features diffusion property; where the first term
of the truncation error is an odd-order spatial derivative, the error features dispersion
property.
(4) Second-order upwind scheme
The second-order upwind scheme can be employed to overcome the shortcomings
of the first-order upwind scheme while taking advantage of the benefit of the upwind
scheme.
For the evenly divided grid shown in Fig. 2.4, the second-order upwind scheme
is defined as
u
∂ϕ
∂ x
≈
u i
2x
(3ϕ i − 4ϕ i−1 + ϕ i−2 ), u i > 0
(2.29a)