The flow-induced flexoelectric effect for tumbling type nematic liquid crystals has been investigated numerically using the Leslie-Ericksen continuum theory. Both the velocity field and the director field of the tumbling nematics are computed for the shear flows between parallel plates, and the magnitude of flexoelectric polarization is estimated from the computed director field. The magnitude is proportional to the difference of the director angle between the plates. Thus, we set weak anchoring condition to one of the plate surfaces to obtain the time dependent magnitude of flexoelectric polarization. The material constants, such as the Leslie viscosities and the Frank elasticities, of 8CB(4-octyl-4'-cyano biphenyl) are used in this work. Dimensionless numbers Er and Ae are chosen to be computational parameters, and are the ratio of viscous force to elastic force and the ratio of the anchoring force to elastic force. For high Ae, the director at the plate can not rotate, and the resulting flexoelectric polarization is small. On the other hand, in the case of low Ae, the director at the plate surface can easily rotate prevailing against the anchoring force and the flexoelectric polarization becomes high. Above a certain critical Er, the director escapes from the shear plane, so called out-of-plane behavior. The magnitude of the flexoelectric polarization becomes maximum when the director at the plate surface is perpendicular to the director at the other side of the plates. In the case of Er=500 and Ae=300, the flexoelectric polarization shows decreasing pulsed profile, and finally reaches zero. As Ae increase, because of high anchoring strength, the director field rapidly reaches its steady state, and the rate of decrease in the flexoelectric polarization becomes higher. The flexoelectric polarization also strongly depends on Er. It is found that the effect of Er on the flexoelectric polarization is opposite to that of Ae, and thus the increase of Er and the decrease of Ae exhibit the similar effect. Finally, the new application of liquid crystalline materials using the dynamic flexoelectric effect will be proposed.