[vvqf]

In OpenFoam, the term (U . Ñ)U in N-S equation is expressed as div(phi,U),where phi is surfaceScalarField, and phi = linearInterpolate(Uo) & mesh.Sf(). I don't understand this.

I think it can also expressed as U . ÑU, (inner product),which in OpenFoam is U & fvc::grad(U)).(why not in this way?)

in OpenFoam, div(phi,U) is said to mean Ñ . (UU), and flux f=U
see User Guide U-110

createPhi.H
#ifndef createPhi_H
#define createPhi_H

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

Info<< "Reading/calculating face flux field phi\n" << endl;

surfaceScalarField phi
(
IOobject
(
"phi",
runTime.timeName(),
mesh,
IOobject::READ_IF_PRESENT,
IOobject::AUTO_WRITE
),
linearInterpolate(U) & mesh.Sf()
);

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

#endif