Abstract: Accurate prediction of molten core debris discharge from a failed reactor pressure vessel lower head is essential for evaluating fuel-coolant interaction, steam-explosion risk, debris-bed formation, and molten core-concrete interaction during a severe accident. Existing integral severe accident analysis codes commonly estimate the discharge velocity using a steady-state Bernoulli equation. This treatment neglects the finite time required to accelerate a large, initially stagnant melt inventory and therefore produces a nonphysical velocity jump immediately after breach formation. To resolve this limitation, this study develops a lumped-parameter numerical method based on the transient Bernoulli principle. The molten core debris is treated as an incompressible fluid, and the fluid-inertia term is retained explicitly in the momentum equation. By combining the transient momentum equation with mass continuity, the governing relation is transformed into a nonlinear first-order ordinary differential equation for the breach velocity. A geometry-dependent integral of the reciprocal flow area is introduced to represent the inertial weighting along the discharge path. Flow-area and liquid-level relations are established for both truncated-cone and spherical-cap lower-head configurations. Distributed and local hydraulic losses are included, together with a remaining-volume-dependent wall-retention correction that represents the rapid increase in resistance at low melt levels. The breach velocity is integrated using the classical fourth-order Runge-Kutta method, while the geometric inertia integral is evaluated using the trapezoidal rule. The remaining melt volume, liquid level, free-surface area, and resistance parameters are updated during discharge. A steady-state Bernoulli model using the same geometry and resistance assumptions is also calculated as a baseline. The method is assessed against the FARO L-14 reference results for the gravity-driven release of 125 kg of UO₂/ZrO₂ melt. The steady-state model predicts an instantaneous maximum velocity at the beginning of release, whereas the transient model starts from zero velocity and reproduces a smooth inertia-controlled acceleration. The predicted velocity rises progressively during approximately the first 0.35 s, reaches a maximum as the decreasing hydrostatic head and increasing resistance become dominant, and then decays nonlinearly. After approximately 0.78 s, the wall-retention correction increases rapidly as the remaining volume enters the low-level regime, causing the discharge velocity to decrease smoothly to zero. The calculated transient evolution agrees well with the FARO reference trend and avoids the nonphysical initial discontinuity of the steady-state solution. The proposed method retains the computational efficiency required by integral severe accident codes while providing a more physically consistent transient boundary condition for subsequent ex-vessel phenomena.