Device for the layered manufacturing of a three-dimensional product from successive powder layers by means of an additive manufacturing technique, wherein this product has at least one recess, wherein this device has a building chamber with a building tube and a building platform which can be moved in a vertical direction in relation to each other, wherein the building platform extends in the building tube with an outer edge connecting to a tube wall of this building tube, wherein a horizontal building surface must extend above the building platform in the building tube, wherein further a powder distributor is provided for applying successive powder layers to said building surface, a radiation source for generating an energetic beam of rays, and scanning means cooperating with the radiation source to make the beam of rays strike said successive powder layers and move across these powder layers in order to form successive interconnected layers of said product, wherein at least one insert is provided which extends vertically, at least partially, through a recess in the building platform provided to that end, so that the building platform and the insert can be moved vertically in relation to each other, wherein the insert substantially fits to the interior of the recess in the building platform, characterized in that this device comprises at least a first gas stream element with at least a first nozzle that opens at the top of the insert, while at least a second gas stream element is provided with at least a second nozzle, in order to allow a gas stream to be generated above said building surface between said first gas stream element and said second gas stream element in order to discharge flue gases, splashes and/or dust particles produced in the course of the manufacture of said product. wherein this product has at least one recess, wherein this device has a building chamber with a building tube and a building platform which can be moved in a vertical direction in relation to each other, wherein the building platform extends in the building tube with an outer edge connecting to a tube wall of this building tube, wherein a horizontal building surface must extend above the building platform in the building tube, wherein further a powder distributor is provided for applying successive powder layers to said building surface, a radiation source for generating an energetic beam of rays, and scanning means cooperating with the radiation source to make the beam of rays strike said successive powder layers and move across these powder layers in order to form successive interconnected layers of said product, wherein at least one insert is provided which extends vertically, at least partially, through a recess in the building platform provided to that end, so that the building platform and the insert can be moved vertically in relation to each other, wherein the insert substantially fits to the interior of the recess in the building platform, characterized in that this device comprises at least a first gas stream element with at least a first nozzle that opens at the top of the insert, while at least a second gas stream element is provided with at least a second nozzle, in order to allow a gas stream to be generated above said building surface between said first gas stream element and said second gas stream element in order to discharge flue gases, splashes and/or dust particles produced in the course of the manufacture of said product.