An engine cooling apparatus comprising: an engine having a cylinder block having a cooling medium passageway in the periphery of a cylinder and a cylinder head, the engine cooling apparatus further comprising: cooling capacity control means comprising a flow rate control valve and/or partial flow rate control valves (61, 62, 63, 64) capable of controlling a cooling capacity of the cylinder head; switching valves (21, 22) capable of changing the flow direction of the cooling medium in the cooling medium passageway, characterized in that the cooling medium passageway comprises a recess-projection portion capable of changing a thermal conductivity to a cooling medium in accordance with a change of a flow direction of the cooling medium, wherein the direction of the cooling medium in the cooling medium passageway is changed by the switching valves (21, 22) between an axial direction of the cylinder and a circumferential direction of the cylinder having a higher thermal conductivity due to the recess-projection portion; and the engine cooling apparatus further comprises control means for performing a control of suppressing the cooling capacity of the cylinder head by controlling the cooling capacity control means so as to close the flow rate control valve or partial flow rate control valves (14, 61, 62, 63, 64) and changing the flow direction of the cooling medium in the cooling medium passageway to coincide with the circumferential direction of the cylinder by controlling the switching valves (21, 22) when the operating conditions of the engine are low-speed, high-load conditions, wherein the recess-projection portion is bent in a wavy form in which faces of the recess-projection portion appear when viewed in the axial direction of the cylinder and the faces do not appear when viewed in the circumferential direction of the cylinder. an engine having a cylinder block having a cooling medium passageway in the periphery of a cylinder and a cylinder head, the engine cooling apparatus further comprising: cooling capacity control means comprising a flow rate control valve and/or partial flow rate control valves (61, 62, 63, 64) capable of controlling a cooling capacity of the cylinder head; switching valves (21, 22) capable of changing the flow direction of the cooling medium in the cooling medium passageway, characterized in that the cooling medium passageway comprises a recess-projection portion capable of changing a thermal conductivity to a cooling medium in accordance with a change of a flow direction of the cooling medium, wherein the direction of the cooling medium in the cooling medium passageway is changed by the switching valves (21, 22) between an axial direction of the cylinder and a circumferential direction of the cylinder having a higher thermal conductivity due to the recess-projection portion; and the engine cooling apparatus further comprises control means for performing a control of suppressing the cooling capacity of the cylinder head by controlling the cooling capacity control means so as to close the flow rate control valve or partial flow rate control valves (14, 61, 62, 63, 64) and changing the flow direction of the cooling medium in the cooling medium passageway to coincide with the circumferential direction of the cylinder by controlling the switching valves (21, 22) when the operating conditions of the engine are low-speed, high-load conditions, wherein the recess-projection portion is bent in a wavy form in which faces of the recess-projection portion appear when viewed in the axial direction of the cylinder and the faces do not appear when viewed in the circumferential direction of the cylinder. cooling capacity control means comprising a flow rate control valve and/or partial flow rate control valves (61, 62, 63, 64) capable of controlling a cooling capacity of the cylinder head; switching valves (21, 22) capable of changing the flow direction of the cooling medium in the cooling medium passageway, characterized in that the cooling medium passageway comprises a recess-projection portion capable of changing a thermal conductivity to a cooling medium in accordance with a change of a flow direction of the cooling medium, wherein the direction of the cooling medium in the cooling medium passageway is changed by the switching valves (21, 22) between an axial direction of the cylinder and a circumferential direction of the cylinder having a higher thermal conductivity due to the recess-projection portion; and the engine cooling apparatus further comprises control means for performing a control of suppressing the cooling capacity of the cylinder head by controlling the cooling capacity control means so as to close the flow rate control valve or partial flow rate control valves (14, 61, 62, 63, 64) and changing the flow direction of the cooling medium in the cooling medium passageway to coincide with the circumferential direction of the cylinder by controlling the switching valves (21, 22) when the operating conditions of the engine are low-speed, high-load conditions, wherein the recess-projection portion is bent in a wavy form in which faces of the recess-projection portion appear when viewed in the axial direction of the cylinder and the faces do not appear when viewed in the circumferential direction of the cylinder.