= Solution
For constant <phase> <density>, let $u_1$ be the intrinsic mean axial speed and define
$$
\boxed{D_1=\frac{\int_\Sigma\chi_1v_x^2\,dA}{A\alpha u_1^2}.}
$$
The profile coefficient measures the momentum <flux> relative to a uniform <velocity> profile with the same mean. Cauchy-Schwarz gives $D_1\geq1$ where the mean is nonzero, with equality for a uniform <phase> profile. At zero mean it is safer to retain the <second moment> itself rather than divide by $u_1^2$.
In the stated common-<pressure>, no-extra-force closure, the averaged conservative momentum equation is
$$
\rho_1[\partial_t(\alpha u_1)+\partial_x(\alpha D_1u_1^2)]+\alpha p_x=0.
$$
For constant $D_1$, expand its derivatives and use $\alpha_t+u_1\alpha_x+\alpha(u_1)_x=0$. The terms combine as
$$
\partial_t(\alpha u_1)+\partial_x(\alpha D_1u_1^2)=\alpha(u_1)_t+(2D_1-1)\alpha u_1(u_1)_x+(D_1-1)u_1^2\alpha_x.
$$
Therefore
$$
\boxed{\rho_1\{\alpha(u_1)_t+(2D_1-1)\alpha u_1(u_1)_x+(D_1-1)u_1^2\alpha_x\}+\alpha p_x=0.}
$$
This is the <phase momentum profile coefficient> correction. It accounts for the <second moment> of <velocity>, not merely a rescaling of the convective speed. A variable coefficient would also produce $\rho_1\alpha u_1^2(D_1)_x$.
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