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P. Liu
transonic flow is a mixture of subsonic and supersonic regions. The boundary
between the subsonic region and the supersonic region is the sonic line.
Before the solution, the position of the sonic line is unknown, so it is necessary to solve the mixed partial differential equation, which brings difficulties
to the theoretical analysis and numerical calculation of transonic flow. Any
small disturbance in the airflow usually propagates at the local sound velocity.
In the transonic flow, most of the flow velocity is close to the sound velocity,
which is similar to the propagation velocity of the above disturbance, so the
disturbance is mainly concentrated in the direction almost perpendicular to
the direction of the incoming flow. Therefore, in the wind tunnel experiment, the disturbance from the surface of the model will directly reflect the
model from the wall of the wind tunnel, or even reflect back and forth many
times. This kind of serious wall interference makes the experimental study of
transonic flow very difficult. In the numerical calculation of transonic flow,
solving the shape and position of the sonic line is an important subject. The
closer the Mach number of the incoming flow is to 1, the larger the area of the
flow near the speed of sound is. The small difference in velocity in the flow
field will cause great changes in the position and shape of the sonic line. The
change in the sonic line has a direct impact on the calculation format of the
flow field. If there is a slight deviation in the calculation of the sonic line and
the ultimate calculation results will be directly affected, which brings many
difficulties to the numerical calculation of transonic flow. In 1971, the American scientists E. M. Muman and J. D. Cole first used the mixed difference
scheme, and successfully solved the steady small disturbance velocity potential equation using the relaxation method. The mixed difference scheme is to
use the central difference scheme in the subsonic region, and the conditions
on all the adjacent nodes will affect the calculation points. In the supersonic
region, the upwind scheme is used, because the upstream upwind node is just
the dependent region of the hyperbolic wave equation. As shown in Fig. 4.15,
the numerical simulation of transonic flow around airfoil is carried out.
4.3.3 Numerical Solution of Supersonic Flow
In supersonic flow, the main problem is how to deal with the shock wave. In
the flow field with the shock wave, the characteristic scale of flow structure
in different regions is very different. The characteristic scale of inviscid shock
wave thickness is zero, while the characteristic scale of the flow field is limited.
The flow parameters change discontinuously when passing through the shock
wave, which brings great difficulty to the calculation of simulated shock wave.
P. Liu
transonic flow is a mixture of subsonic and supersonic regions. The boundary
between the subsonic region and the supersonic region is the sonic line.
Before the solution, the position of the sonic line is unknown, so it is necessary to solve the mixed partial differential equation, which brings difficulties
to the theoretical analysis and numerical calculation of transonic flow. Any
small disturbance in the airflow usually propagates at the local sound velocity.
In the transonic flow, most of the flow velocity is close to the sound velocity,
which is similar to the propagation velocity of the above disturbance, so the
disturbance is mainly concentrated in the direction almost perpendicular to
the direction of the incoming flow. Therefore, in the wind tunnel experiment, the disturbance from the surface of the model will directly reflect the
model from the wall of the wind tunnel, or even reflect back and forth many
times. This kind of serious wall interference makes the experimental study of
transonic flow very difficult. In the numerical calculation of transonic flow,
solving the shape and position of the sonic line is an important subject. The
closer the Mach number of the incoming flow is to 1, the larger the area of the
flow near the speed of sound is. The small difference in velocity in the flow
field will cause great changes in the position and shape of the sonic line. The
change in the sonic line has a direct impact on the calculation format of the
flow field. If there is a slight deviation in the calculation of the sonic line and
the ultimate calculation results will be directly affected, which brings many
difficulties to the numerical calculation of transonic flow. In 1971, the American scientists E. M. Muman and J. D. Cole first used the mixed difference
scheme, and successfully solved the steady small disturbance velocity potential equation using the relaxation method. The mixed difference scheme is to
use the central difference scheme in the subsonic region, and the conditions
on all the adjacent nodes will affect the calculation points. In the supersonic
region, the upwind scheme is used, because the upstream upwind node is just
the dependent region of the hyperbolic wave equation. As shown in Fig. 4.15,
the numerical simulation of transonic flow around airfoil is carried out.
4.3.3 Numerical Solution of Supersonic Flow
In supersonic flow, the main problem is how to deal with the shock wave. In
the flow field with the shock wave, the characteristic scale of flow structure
in different regions is very different. The characteristic scale of inviscid shock
wave thickness is zero, while the characteristic scale of the flow field is limited.
The flow parameters change discontinuously when passing through the shock
wave, which brings great difficulty to the calculation of simulated shock wave.