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A high molecular polymer solution with viscoelasticity has the effect of reducing frictional drag, which is quite practical for energy saving. Effective simulations of viscoelastic flows in a pipeline with a high Reynolds number is realized by incorporating the constitutive equation of viscoelasticity into the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>−</mo><mi>ε</mi><mo>−</mo><mover accent="true"><mrow><msup><msup><mi>v</mi><mo>′</mo></msup><mn>2</mn></msup></mrow><mo stretchy="true">¯</mo></mover><mo>−</mo><mi>f</mi></mrow></semantics></math></inline-formula> turbulence model. The Finitely Extensive Nonlinear Elastic Peterlin (FENE-P) model is employed for characterizing the viscoelasticity. The drag reduction of fully developed viscoelastic pipe flow with a fixed mass flow rate is studied. Different from increasing the center velocity and without changing the velocity near the wall at a fixed pressure drop rate, the addition of a polymer reduces the velocity near the wall and increases the velocity at the center of the pipe and makes the flow tend to be a laminar flow. Decreasing the solvent viscosity ratio or increasing the maximum extensibility or the Weissenberg number can effectively reduce the turbulence intensity and the wall friction. Under the premise of ensuring calculation accuracy, this Reynolds-averaged simulation method for viscoelastic flow has significant advantages in both computational cost and accuracy, which is promising for drag reduction simulation and practical engineering applications.